Coordination in Supply Chains When Demand ... - Semantic Scholar

Coordination in Supply Chains When Demand Forecasts Are Not Common Knowledge: Theory and Experiment Kyle Hyndman Southern Methodist University, [email protected], http://faculty.smu.edu/hyndman

Santiago Kraiselburd Zaragoza Logistics Center & INCAE Business School, [email protected]

Noel Watson Zaragoza Logistics Center, [email protected]

We study the coordination problem of a two-firm supply chain in which firms simultaneously choose a capacity before demand is realized. We focus on the role that (a lack of) common knowledge of demand forecasts has on firms’ ability to align their capacity decisions. When forecasts are not common knowledge, there are at most two equilibria: a complete coordination failure or a monotone equilibrium in which firms earn positive profits. The former equilibrium always exists, while the latter exists only when the marginal cost of capacity is sufficiently low, though capacities are misaligned with probability 1 and profits are lower than in the efficient equilibrium of the common knowledge game. We also show that both truthful information sharing and pre-play communication have an equilibrium with higher profits. We then test the model’s predictions experimentally. Contrary to our theoretical predictions, we show that non-common knowledge of demand forecasts does not have a consistently negative affect on firm profits, though capacities are more misaligned. We also show that pre-play communication is just as effective at increasing profits and improving alignment as is truthful information sharing. Moreover, we also document that “honesty is the best policy” when it comes to communicating private information. Key words : Communication, Coordination, Global Games, Supply Chains

1.

Introduction

There is an extensive literature in operations management (OM) that studies coordination either between firms or across different functional units within a vertically integrated firm. In this literature, to achieve coordination, two conditions are necessary: (a) that the individual player’s decisions are aligned, and (b) that alignment occurs at a point that maximizes system’s profits. Unlike much of the literature, we focus on alignment of activities across multiple decision-makers, without it necessarily also maximizing joint profits. Our study is not the first to focus on alignment. Oliva and Watson (2010) argue for alignment as an independent activity which brings unique benefits to the organization/supply-chain. Additionally, Kraiselburd and Watson (2010), focus on generating 1

2

Hyndman, Kraiselburd and Watson: Coordination in Supply Chains

hypotheses about alignment in different organizational and supply chain settings. Our study of coordination through alignment is part of a recent trend of viewing certain OM themes through increasingly more complex behavioral lenses. This allows us to isolate the behavioral factors that may influence decision making and de-emphasize the supply chain’s physical/structural factors that can also influence overall performance but are more well-studied in the literature. In this paper, we define a two-player game where the two players simultaneously invest in capacity in order to meet demand, and sales are equal to the minimum of the two players’ capacities and realized demand — a variation on the newsvendor model. In this setup, if one player plans for a given capacity to meet their perceived demand, the other player would prefer not to invest in any more capacity, since the effective capacity is given by the constraining capacity. We pay particular attention to the role of information about demand — especially the distinction between whether or not information is common knowledge — on equilibrium behavior. A leading example that fits nicely in this setting is that of an assembly system in which two suppliers provide components for a final good. In such situations, sales are limited by either demand or the constraining capacity. A number of real world situations fit this scenario. For example, numerous reports have circulated about bottlenecks in the production of the Toyota Prius due to limited supplies of battery packs and inverters.1 Similarly, production of Boeing’s 787 airplane has been delayed several times due to bottlenecks at some key component suppliers.2 Tomlin (2003) cites excerpts from Palm Inc.’s financial report that caution about potential inadequate capacity problems from third party suppliers given uncertainties in demand for handhelds. These examples highlight the difficulty that firms have in aligning production in the face of uncertain demand, and show how misalignment can lead to severe delays and lost sales opportunities. There are other situations where our model applies. For example, consider an integrated firm introducing a new product. In this case, the manufacturing group must invest in capacity to actually produce the product, and the sales group must invest in its sales force or in a marketing campaign to inform potential buyers of the product’s existence. An important feature of these examples is that of strategic complementarities: so long as capacity is less than demand, higher capacity choices lead to higher profits, provided both firms choose a higher capacity. Of course, demand is unobserved before capacities are chosen, and this uncertainty adds an additional layer of complexity. It turns out that a crucial issue is whether or not all parties involved have common information or whether each firm has its own private information about 1

For example, see, “Parts shortages crimp Toyota’s Prius output,” (Automotive News; November 3, 2009).

2

For example, see, “A Dream Interrupted at Boeing,” (New York Times, September 6, 2009).


3

demand. In the former case, there are multiple, Pareto rankable equilibria of the game sketched above (Tomlin 2003). Therefore, an important unanswered question is, which equilibrium can we expect firms to play? Numerous experimental studies have shown that such games often lead to coordination failures.3 Our paper takes a first step at examining this issue by showing that the multiplicity of equilibria leads to coordination problems and sub-optimal earnings. Matters are different when each firm has private information. To model this, we borrow from the economics literature on global games, first studied by Carlsson and van Damme (1993). That is, we assume that demand is a random draw from a prior distribution, which is commonly known to both firms. While demand is unobserved, each firm receives an independent, private signal which is correlated with demand. In the present context, one can view each firm’s private signal as the result of its demand forecast. To the extent that each firm has its own forecasting method or makes use of different data, it is very likely that their forecasts will be different.4 Assuming that players are symmetric, in contrast to the common knowledge game, we show that there are at most two equilibria. There is always the complete coordination failure equilibrium in which both players choose zero capacity regardless of their signal. However, provided that the marginal cost of capacity is below a threshold, there is an equilibrium in which players’ capacity choice functions are increasing in their signal. In this equilibrium, capacities and profits are lower than in the Pareto efficient equilibrium of the common knowledge game. Moreover, since players receive independent signals, capacities are necessarily misaligned, which lowers aggregate profits because misalignment means that at least one player over-invested in capacity. We next turn to a study of mechanisms that can lead to improved coordination. We first focus on truthful information sharing in which, say, firms have reached an agreement to share their demand forecasts with each other. Not surprisingly, truthful information sharing may lead to an increase in profits relative to both the common knowledge and non-common knowledge games. Under truthful information sharing, both players receive two signals about demand, which leads to more precise information. We show that the efficient equilibrium of this game leads to profits which are higher than the efficient equilibrium of the common knowledge game. Of course, such a mechanism requires institutions to ensure that the information shared is, in fact, truthful. Since these such institutions may be impossible or prohibitively costly, we also consider a game in which pre-play communication between players is allowed. Here we show a surprising 3

See, e.g., Hyndman, Terracol, and Vaksmann (2009), Brandts and Cooper (2006) and van Huyck, Battalio, and Beil (1990), among others. 4

For example, it may be that the manufacturer also observes demand for a related product, while at the same time the sales group may base their forecast on focus group studies that it does not share with the manufacturer.

4


result; namely, firms may be able to achieve the same benefits through pre-play communication only. That is, there exists an equilibrium in which (i) firms truthfully report their signals and (ii) firms coordinate on the Pareto efficient equilibrium of the game with truthful information sharing. The logic behind this result is straightforward. Since firms expect to coordinate on the efficient equilibrium in the post-communication subgame, and since lies that have been discovered (which happens with positive probability) are punished, they can only suffer by lying about their signal. This result shows that, while pre-play communication may be as effective as truthful information sharing, it need not be so. If firms do not expect to coordinate on the efficient equilibrium, we show by means of an example that they may have an incentive to inflate their message The theoretical results have a number of testable implications. Because of multiple equilibria, there are also some areas where theory does not make a clear prediction. Therefore, we conduct a series of experiments in order to see how closely behaviour matches the theory, and also to shed light on those areas where the theory is ambiguous. Our experiment was conducted with two main questions in mind. First, what is the difference in actual behaviour when demand forecasts are common vs. non-common knowledge? Second, are there benefits from information sharing, and if so, can these benefits be achieved through pre-play communication or do they only accrue when mechanisms for truthful information revelation are in place? To study these questions, our experiment has four information treatments: (i) the common knowledge game (cp), (ii) the non-common knowledge game (ncp), (iii) the common knowledge game with two signals (cp-2s) and (iv) the non-common knowledge game with communication (ms). The cp and ncp treatments allow us to investigate the first question, while the cp-2s and ms treatments allow us to investigate the second question. For each treatment, we conduct sessions with different sets of parameters in order to examine the comparative statics of each treatment. Regarding the first question, recall that the efficient equilibrium of the common knowledge game leads to higher expected profits than in the non-common knowledge game. That being said, the common knowledge game has a continuum of equilibria, while the non-common knowledge game has at most two equilibria. Therefore, it is possible that earnings are lower under common knowledge. Indeed, our results show that average profits in the common knowledge game are not consistently higher than the non-common knowledge game. While profits are not consistently higher in the cp game, it is generally true that capacities are better-aligned. We also show that subjects appear to be risk averse. While choices are below the theoretical prediction, the difference is increasing in the signal (where the potential to be undercut is greater). Additionally, “anchoring and insufficient adjustment” does not explain comparative statics on the


5

newsvendor critical fractile. While subjects under-adjust when it is high, they over-adjust when it is low. One striking inconsistency with our theory is that in the ncp game we do not see the predicted complete coordination failure when the marginal cost of capacity is high. At the same time, while learning leads to better alignment and higher profits in the cp treatment, the same does not hold for the ncp treatment. In fact, there is a downward trend in profits in the final 10 rounds of the ncp game where the complete coordination failure is predicted. We next turn our attention to the two mechanisms with the potential to improve coordination. While our theoretical result suggests that ms may be equivalent to cp-2s, we did not expect this to be so. Truthful revelation is only an equilibrium when subjects expect to play the efficient equilibrium of the cp-2s game. Given that we saw efficiency gaps of as much as 15% in our cp and ncp treatments, we expected that this would lead subjects to inflate their signals, making communication ineffective. Surprisingly, however, while only 25% of messages were truthful, the ms treatment actually leads to marginally higher average profits than the cp-2s treatment. Given that subjects lie about their messages, we are left wondering why communication lead to higher profits. We believe that there are at least three reasons for this. First, messages are an increasing function of one’s estimate. Therefore, even though subjects are lying, this function can be inverted to (noisily) reveal the true estimate. Second, one’s own choice is positively correlated with the message that they send. This provides subjects with an opportunity to make inferences about their match’s eventual choice. Third, if a subject receives a message which is too far from her own signal, then she chooses a significantly lower capacity. This is precisely the kind of off-theequilibrium-path punishments that sustain the truth-telling equilibrium of the ms game. It appears that such punishments are meted out and may ensure that people are not too dishonest. Finally, our results show that honesty is the best policy when it comes to sending a message. In particular, profits are decreasing in the difference between one’s message and ones’ signal. Profits are also decreasing in the difference between the message received and that player’s signal. The paper proceeds as follows. Section 2 reviews the relevant literature. Section 3 describes the model, Section 4 contains our theoretical results and Section 5 discusses our experimental design. Section 6 discusses the results of our cp and ncp treatments, while Section 7 examines the results of our cp-2s and ms treatments. Finally, Section 8 provides some concluding remarks.

2.

Literature Review

A number of authors have studied coordination between firms in an OM context (see Cachon (2003) for a review). Among the many contributions, two relevant examples are Shapiro (1977)

6


and Lee and Whang (1999), who study coordination between different functions within a vertically integrated firm. More recently, Chao, Seshadri, and Pinedo (2008) examine optimal capacity in coordinated supply chains in which the manufacturer must eventually fulfill the retailer’s order and the retailer employs a periodic review system. Our focus on the joint capacity decision by firms in a supply chain puts us closer in spirit to Tomlin (2003), who studies coordination when both firms have identical beliefs about demand. However, above and beyond the fact that our paper focuses on experimentally testing predictions, there are two important differences. First, Tomlin (2003) largely ignores the presence of multiple equilibria, while this is key to our analysis since we study alignment. Second, Tomlin (2003) only considers the case of common information, while we also study the effect of private information on equilibrium behavior. In contrast to our approach, the OM literature has generally emphasized coordination instead of alignment. The intrafirm literature has either concentrated on approaches for managing the sales force (Chen 2005, Gonik 1978, Lal and Staelin 1986), or considered schemes for coordinating functions so as to achieve the benefits of centralized decision making (Porteus and Whang 1991, Celikbas, Shanthikumar, and Swaminathan 1999, Li and Atkins 2002). More qualitative treatments of intrafirm coordinations include Kahn and Mentzer (1996) and Fawcett and Magnan (2002). The related interfirm literature, which considers the interactions between manufacturers and retailers that are not in the same firm, generally concentrates on finding new incentive schemes to achieve the benefits of centralized decision making (Cachon 2003, Cachon and Lariviere 2001, Tomlin 2003). Our paper also studies the role of information sharing in helping firms achieve a more profitable outcome (Cachon and Fisher 2000, Aviv 2001). Within the collaborative forecasting literature, Miyaoka (2003), Lariviere (2002) and Ozer and Wei (2006) argue that whether the parties reveal truthful information depends on their incentives, and go on to design truth-revealing mechanisms. In Kurtulus and Toktay (2007), parties must decide whether to invest to improve their forecasting before sharing it. Other papers, such as Li and Zhang (2008), Li (2002) and Jain, Sohoni, and Seshadri (2009) study information sharing in which one or more retailers can share their information with a single manufacturer. In contrast, we focus on two-sided information sharing. Our paper also contributes to the growing literature on behavioral OM, which is summarized by Bendoly, Donohue, and Schultz (2006). We briefly touch upon some of the decision biases noted by Schweitzer and Cachon (2000). The paper that is closest to our work is Ozer, Zheng, and Chen (2008). They study one-way communication in a standard newsvendor experiment, while we examine the beneficial effects of two-way communication in an environment where both firms must invest in capacity. In both papers, subjects are free to send untruthful messages.


3.

7

The Model

Consider a two-firm supply chain. We will often refer to one firm as manufacturing, m, and to the other firm as sales, s. Manufacturing supplies sales with a product that sales converts into a final product. For demand to be met, each firm needs to invest in capacity. Denote by Ki , i ∈ {m, s} the capacity of firm i. Assume that the unit cost of capacity is γ > 0 for each firm and that the net revenue per unit sold is πm > 0 for manufacturing and πs > 0 for sales. One could view πm and πs as being derived from a simple transfer pricing scheme. All parameters are common knowledge. The timing of events is as follows. First, firms m and s simultaneously choose their capacities, Km and Ks . Second, demand, x, is realized and sales are given by min{x, Ks , Km }. Therefore, the profits of firm i ∈ {s, m} can be written as: Πi (x, Ks , Km ) = πi min{x, Ks , Km } − γKi . Inspired by the literature on global games in (see, e.g., Carlsson and van Damme (1993)), we provide a specific structure on demand. In particular, suppose that both firms have a diffuse prior about demand, x, over R (i.e., demand is equally likely to be any number on R).5 Prior to choosing capacities, each firm receives a private signal: θi = x + i , where i ∼ U [−η, η] and η > 0 measures the noisiness of the signals.6 We consider two cases. First, we consider the case in which m = s and that this is common knowledge. In this case, the model is a special case of Tomlin (2003), where the distribution of demand, conditional on θ, is U [θ − η, θ +η]. Second, we consider the case in which s and m are independent draws from the distribution U [−η, η]. In this case, the firms do not have a commonly held demand forecast. For example, given

a signal θi received by firm i, this firm believes that the true state is uniformly distributed on [θi − η, θi + η], while firm i’s beliefs about the signal received by player j are not uniform. This is depicted graphically for a specific set of parameters in Figure 1. Note that unconditional on the state, signals are positively correlated. Therefore, if one firm receives a higher signals, then it believes that the other firm likely received a higher signal as well. Before we proceed, some notation is in order. Let F (x|θ) denote the distribution over demand states conditional upon receiving a signal θ, and let f (x|θ) denote the corresponding density function. By assumption, F is uniformly distributed over [θ − η, θ + η]. Additionally, let Gi (θj |x) denote 5

Note that the prior on demand has infinite support, and is termed an improper prior. This does not pose any technical problems for us since we are only concerned with conditional beliefs (see Hartigan (1983)). 6

The uniform distribution is for analytical convenience. Qualitatively identical results go through if i ∼ N (0, σ 2 ).

8


Figure 1

Beliefs about the state and the other player’s signal 0.06

Beliefs about demand

0.055

0.05

0.045

Beliefs about

opponent's signal

Density

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0

80

85

90

95

100

105

110

115

120

Support of demand and possible signals

the distribution of player i’s beliefs about the signal received by player j conditional on the true state. Given our setup, Gi (θj |x) is uniform over [x − η, x + η]. Remark 1 (Prior Beliefs About Demand). The assumption that the prior beliefs about demand are diffuse over R is unrealistic, but is done for analytical convenience. If priors were diffuse over R+ , then if θi ∈ [−η, η), firm i has more precise information about demand than if θi ≥ η. Moreover, over the range [−η, η), as θi increases, firm i’s information becomes less accurate. While one can handle this, it makes the analysis more complicated without adding any additional insight.

4. 4.1.

Equilibrium Characterization The Common Knowledge Game

We can write the expected profit function of each firm, having received the common signal θ, as: ¯ s (θ, Ks , Km ) = πs Π

Z

¯ m (θ, Ks , Km ) = πm Π

θ+η

min{x, Ks , Km }f (x|θ)dx − γKs Zθ−η θ+η

min{x, Ks , Km }f (x|θ)dx − γKm . θ−η

Recall that the two firms simultaneously choose capacity, Ki ≥ 0 to maximize their expected profits. Define s∗ =

πs −γ πs

and m∗ =

πm −γ , πm

where s∗ and m∗ are the traditional newsvendor critical fractiles

for sales and manufacturing. Then we have the following result: Proposition 0. For θ ≤ max{η(1 − 2s∗ ), η(1 − 2m∗ )}, there is a unique equilibrium in which both firms choose Ki (θ) = 0. For θ > max{η(1 − 2s∗ ), η(1 − 2m∗ )}, there are multiple equilibria of the capacity choice game. In the Pareto efficient equilibrium outcome, both firms choose capacity Ks (θ) = Km (θ) = min{F −1 (s∗ |θ), F −1 (m∗ |θ)}. Proof See Appendix.

9


Notice that if both firms are symmetric, then the Pareto efficient equilibrium choice function corresponds to the single-player newsvendor solution. Although it is common to assume that firms can coordinate on the Pareto efficient choice rule, the presence of multiple equilibria means that there is no guarantee that firms will be able to do so. 4.2.

The Non-Common Knowledge Game

We turn now to the case in which each firm receives a private signal θi = x + i . This assumption is meant to capture the idea that different firms may have access to different information or methodologies in deriving their demand forecast. We characterize an equilibrium in which both firms employ monotonic strategies, Ki (θi ) < θi + η, where Ki0 > 0 for all θ ≥ θ. It is quite natural to focus on such strategies since they reduce the perceived complexity of the game. Beyond that, monotone strategies are quite robust. For example, if firm j uses a monotone strategy, it will generally be a best response for firm i to play a monotone strategy as well. The intuition is straightforward. If firm i’s signal increases slightly, then its expectation of demand increases, as does its belief about firm j’s signal. Since firm j is playing a monotone strategy, firm i therefore, expects firm j’s capacity choice to increase. Thus, it will be optimal for firm i to increase its capacity as well. In general, the expected profit function of the sales firm is: ¯ s (θs , Ks , Km (·)) = πs Π =

Z

πs 4η 2

θs +η

Z

x+η

min{x, Ks , Km (θm )}gs (θm |x)dθm f (x|θs )dx − γKs θs −η Z θs +η Zx−η x+η

min{x, Ks , Km (θm )}dθm dx − γKs , θs −η

x−η

where Km (θm ) is the strategy employed by the manufacturing firm as a function of its observed signal and x is the realization of demand over which we integrate. The second line follows because both gs (θm |x) and f (x|θs ) are uniform densities with a support of length 2η. Similarly, for the manufacturing firm, we have that: ¯ m (θm , Km , Ks (·)) = πm Π 4η 2

Z

θm +η

Z

x+η

min{x, Ks (θs ), Km }dθs dx − γKm . θm −η

x−η

In what follows, we will assume that the firms are symmetric, meaning that π ≡ πs = πm . 4.2.1.

Main Characterization. We now provide a complete characterization of equilibrium

behavior in the non-common knowledge game. Proposition 1. For all γ ∈ (0, π), there is an equilibrium such that K(θ) = 0 for all θ. If γ >

10


γ¯ ≡ π2 , then this is the unique equilibrium. If γ ≤ γ¯ , there is a symmetric equilibrium in monotone strategies. In this equilibrium, firms’ capacity choices are given by:  q  0, if θ ≤ η − 2η 1 − 2γ π q q K(θ) =  θ − η + 2η 1 − 2γ , if θ > η − 2η 1 − 2γ . π

π

If K(θ) > 0, then it is less than the capacity choice in the efficient equilibrium of the cp game. Furthermore, within the class of monotone strategies, the equilibrium is unique. Proof See Appendix.

That the complete coordination failure exists is straightforward. It is trivially optimal to choose a Ki (θ) = 0 for all θ if Kj (θj ) = 0 for all θ. That this is the unique equilibrium when γ >

π 2

is

also not difficult. In equilibrium, each firm expects, on average, that the other firm will have a lower signal than them half of the time. Since the cost of capacity is high relative to the marginal revenue of sales, it makes sense for the firm to lower its capacity. However, both firms have the same incentives, which means that each firm’s negative expectations are mutually reinforced until a complete coordination failure occurs. This intuition also shows us that there cannot be an asymmetric equilibrium when γ >

π 2

since

then Ki (θ) 6= Kj (θ). In this situation, for the firm choosing a higher capacity, it would be the case that more than half the time its opponent will choose a lower capacity, giving the firm a strong incentive to lower its capacity. The remainder of our analysis assumes that γ ≤ γ¯ so that we may be assured of a symmetric monotone equilibrium. Observe that in the efficient equilibrium of the cp game as well as in the monotone equilibrium of the ncp game, the capacity choice function is affine in signal received, with a slope of 1. Furthermore, observe that the critical signal, above which firms choose a strictly positive capacity, is strictly higher in the ncp game. That is, θncp > θcp . Taken together, these two observations imply that K ncp (θ) < K ncp (θ). That is, for the same signal, firms in the cp game make higher capacity choices than firms in the ncp game. Of course, this is due to the fact that, because firms use monotone strategies and signals are conditionally independent, capacity choices will be misaligned with probability 1. Because of this, firms will be more cautious in their capacity choice when demand forecasts are not common knowledge. Therefore, we also have the following: Corollary 1. Expected profits in the non-common knowledge game are strictly less than expected profits in the Pareto efficient equilibrium of the common knowledge game.


11

Remark 2 (Prior Beliefs About Demand cont’d ). As was mentioned above, we assumed that firms have a diffuse prior over R about demand. A more realistic assumption is that priors are √ diffuse over R+ . In this case, we are able to show that for γ ≤ γ¯ ≡ ( 2 − 1)π, there is a symmetric √ p monotone equilibrium such that K 0 (θ) = θ − η +2 η η − (γ/π) min{2η, θ + η }. Furthermore, when γ > π2 , exactly the same logic tells us that there must be a full coordination failure, at least for θ > η. However, because of the greater precision of signals, we are unable to get a complete characterization of behavior for γ > γ¯ and θ < η. 4.2.2.

Information Sharing and Communication By Firms. Corollary 1 shows that

expected profits in the ncp game are lower than the expected profits in the Pareto efficient equilibrium of the cp game. Proposition 1 demonstrates the possibility of a complete coordination failure if γ > π2 . Therefore, we turn our attention to some ways in which firms may be able to overcome the problems due to a lack of common knowledge. We first focus on information sharing by firms. Suppose that there is a mechanism in which firms receive each other’s signal, in addition to their own. This mechanism accomplishes two things. First, it makes the information that each firm has more precise. Second, it restores common knowledge. We call this game the common knowledge game with two signals (cp-2s). Because information is more precise, it is possible that expected profits are higher in cp-2s; however, this need not be so since there are also multiple equilibria. More formally, let θ` = min{θs , θm } and θh = max{θs , θm }. Then the support of demand is [θh − η, θ` + η] and updated beliefs about demand, conditional on θm and θs , are uniform on that support. Then we have: Proposition 2. There are multiple equilibria of the cp-2s game. The Pareto efficient equilibrium is characterized by K cp-2s (θ` , θh ) = max{s∗ (θ` + η) + (1 − s∗ )(θh − η), 0}. Moreover, expected profits in the Pareto efficient equilibrium are higher than without information sharing. Proof This is a direct application of Proposition 0 upon updating the players’, now common, beliefs about demand.

Of course, it may be difficult or prohibitively costly for firms to set up such a mechanism to convey their private information. A more realistic and cheaper scenario may be one in which, prior to setting capacities, firms engage in pre-play communication. We briefly explore whether this also has the potential to lead to higher expected profits and better alignment of capacities. The timing of the non-common knowledge game with communication (ms) is as follows. First, each firm receives its private signal, θi , for i ∈ {m, s}. Second, firms simultaneously send a message

12


Mi ∈ R to the other firm. Third, firms simultaneously choose capacities, Ki . Finally, demand is realized and sales are given by min{x, Km , Ks }. While “babbling” equilibria always exist in which firms ignore messages and play an equilibrium of the ncp game, we show that there is also an equilibrium in which firms truthfully report their signals and they coordinate on the Pareto efficient equilibrium of the cp-2s game. Formally, Proposition 3. There exists an equilibrium of the ms game in which firm i sends message, Mi = θi , i ∈ {m, s}, and both firms subsequently choose capacity K cp-2s (θ` , θh ). Proof See Appendix.

The intuition for this result is straightforward. Since firms expect to coordinate on the efficient equilibrium in the capacity choice subgame, there is no benefit to lying about their signal (even if firm i could get j to increase its capacity relative to truth-telling, firm i would still choose the same capacity). On the other hand, there are costs associated with lying. First, if it turns out that |Mi − θj | > 2η, then firm j knows that firm i has lied, and will subsequently set Kj = 0 in the capacity choice subgame. Second, even if |Mi − θj | ≤ 2η, if Mi < θi , then firm j will choose a lower-than-optimal capacity, which will lead to lower profits. We conjecture that the equilibrium described in Proposition 3 is the unique truthful equilibrium, though we have been unable to formally prove this. Note that in a truthful equilibrium, capacities are increasing in the message received. Since firms are not at the efficient capacities, they would like to increase their capacity.7 The only way to accomplish this is to make the other firm think demand is higher. Of course, by doing so, the firm takes a risk that it will be found out to have lied, and so will subsequently be punished. However, we believe that there will always be a profitable deviation because the probability of being discovered to have lied is quite small, while there is a first-order benefit due to increased capacity in the likely event that the firm’s lie goes undiscovered. The supplemental notes contains an example to provide further intuition for our conjecture. 4.3.

Discussion of Testable Hypotheses

In order to test our theory, we will divide our discussion of the experiment in two main parts. We will first compare behaviour in the cp and ncp games. Our goal for these two treatments is to identify the practical role that private information has on outcomes. The purpose of our study of behaviour in cp-2s and ms will be to determine whether there are benefits to information sharing, and whether these benefits can be achieved through communication or whether one needs to set up mechanisms in which information is truthfully shared, as in cp-2s. We make the following hypotheses, which will be formally tested in the experiment: 7

That is,

π 2η+θ` −θh

R θ` +η θh −η

min{x, Ki (θ` , θh )}dx − γKi (θ` , θh ) is strictly increasing at Ki (θ` , θh ).


13

¯ Hypothesis 1 (Average Profits (Π)). We expect the following ordering of average profits ¯ ¯ ¯ ¯ across treatments: Π(cp-2s) > Π(ms) > Π(cp) > Π(ncp). Hypothesis 2 (Average Misalignment (d¯ = |K1 − K2 |)). We expect the following ordering ¯ ¯ ¯ ¯ of average misalignment of capacities across treatments: d(cp-2s) < d(ms) < d(cp) < d(ncp). ¯ ¯ Consider first Hypothesis 1, which concerns average profits. That Π(cp) > Π(ncp) follows from Proposition 1. In particular, when γ > π/2, the unique equilibrium in the ncp game is the complete coordination failure. Therefore, despite the multiplicity of equilibria in cp, average profits should be higher. Furthermore, even when γ < π/2, the Pareto efficient equilibrium of cp leads to higher profits than in the efficient (monotone) equilibrium of ncp. ¯ ¯ Next, consider the Π(cp-2s) > Π(ms) part of Hypothesis 1. Proposition 3 suggests that there exists an equilibrium of the ms game in which players truthfully report their signals and coordinate on the efficient equilibrium of the cp-2s game. However, it is our belief that coordinating on this equilibrium will be difficult for players, which opens up the possibility that players will lie and “discount” the messages that they receive; hence profits will be lower in ms. That being said, we do believe that communication in the ms treatment will not be totally without benefit; in particular, we believe that subjects will not lie “too much”, making it possible to extract some information. ¯ ¯ Hence, we predict that Π(ms) > Π(cp). Note that our predictions are based on the presumption that players in the cp-2s and cp treatments will be able to coordinate close to the Pareto efficient equilibrium. To be sure, given the fact that there is a continuum of equilibria in these games, it is certainly possible that they will coordinate on an inefficient equilibrium, or that different players will play according to different equilibria. Therefore, to the extent that some of the inequalities in Hypothesis 1 do not hold, this can be seen as evidence that multiplicity of equilibria inherent in the common knowledge games has practical, and detrimental, consequences for players. Turn now to Hypothesis 2, which relates to the misalignment of capacity choices, d¯ = |K1 − K2 |. ¯ ¯ Considering first the d(cp) < d(ncp) part of the hypothesis, recall that, despite the multiplicity of equilibria in the cp game, in all of them, capacities are perfectly aligned. In contrast, in the monotone equilibrium of the ncp game, which holds when γ < π/2, capacities are misaligned with probability 1. To be sure, when γ > π/2, it is possible that subjects are perfectly aligned on the complete coordination failure; however, we believe that this is extremely unlikely in a two-person ¯ ¯ game. Next turn to our prediction that d(cp-2s) < d(ms). Given that we expect players to lie in the ms treatment, we believe that it will lead to greater misalignment as compared to the cp-2s treatment. At the same time, as was the case with average profits, we expect that messages in the ¯ ¯ ms treatment will be partially informative, leading to our prediction that d(ms) < d(cp).

14


5.

Experimental Design

The experiments were run at the experimental economics laboratory of a public university in the United States. The 192 subjects in this experiment were recruited from undergraduate classes and had no previous experience in our experiments. In each session, after subjects read the instructions they were also read aloud by an experimental administrator. Sessions lasted for between 45 and 90 minutes depending on the treatment, and each subject participated in only one session. A $5.00 show-up fee and subsequent earnings, which averaged about $18.00, were paid in private at the end of the session. Throughout the experiment, we ensured anonymity and effective isolation of subjects in order to minimize any interpersonal influences that could stimulate uniformity of behavior.8 The experiment consists of four main treatments: (i) the common knowledge game (cp), the non-common knowledge game (ncp), (iii) the common knowledge game with two signals game (cp-2s) and (iv) the non-common knowledge game with communication (ms). The basic structure of each treatment was as follows. First, the prior distribution of demand was uniform with support [x, x], where 0 < x < x. While our theoretical results were derived for the case of a diffuse prior, this is not implementable in the lab and was a necessary modification. None of the theoretical results are sensitive to this modification. Second, conditional on the state, x, subjects received a signal θi = x + i , where i ∼ U [−η, η]. Third, the profit function was: πi (Ki , Kj , x) = π min{Ki , Kj , x} − γKi . Within each of these four treatments, we ran multiple sessions in which the parameters x, x, η, π and γ were varied. In all treatments, subjects were randomly re-matched after each round and subjects played the game in their session for either 30 or 40 rounds. 5.1.

Details of Each Treatment

Table 1 summarizes the details of our experiment. A sample of the instructions used can be found in the supplemental notes. The experiment was programmed using z-Tree (Fischbacher 2007). Below, we briefly remind the reader about any important features not captured in Table 1. Note that it will often be convenient to refer to a specific game by the abbreviation cp(π, γ) or ncp(π, γ). 5.1.1.

Common Knowledge Game (cp) In this treatment, for each pair, demand, x, was

drawn from the appropriate distribution in Table 1 and both subjects received the signal θ = x + . That is, both subjects received the same signal about demand and this was common knowledge. 8

Participants’ workstations were isolated by cubicles making it impossible for participants to observe other’s screens or to communicate. We also made sure that all remained silent throughout the session. At the end of a session, participants were paid in private according to the number on their workstation.

15


Table 1

Summary of experiments

Treatment cp

ncp

cp-2s

ms

5.1.2.

π

γ

5 10 10 5 10 10 5 10 10 5 10 10

2 3 6 2 3 6 2 3 6 2 3 6

Prior on Demand U [20, 50] U [100, 400] U [100, 400] U [20, 50] U [100, 400] U [100, 400] U [20, 50] U [100, 400] U [100, 400] U [20, 50] U [100, 400] U [100, 400]

Noisiness of Signals (η) 5 25 25 5 25 25 5 25 25 5 25 25

Number of Rounds 30 40 40 30 40 40 30 40 40 30 40 40

Number of Subjects 20 12 10 18 24 8 22 10 12 22 24 10

Non-Common Knowledge Game (ncp) In this treatment, for each pair, demand,

x, was drawn from the appropriate distribution in Table 1 and each subject, i, received a signal θi = x + i . That is, both subjects received a different signal about demand and it was known that this would be the case. In the games ncp(5, 2) and ncp(10, 3), the monotone equilibrium exists, while in the ncp(10, 6) game, the unique equilibrium is that of the complete coordination failure. 5.1.3.

Common Knowledge Game With Two Signals (cp-2s) In this treatment, for

each pair, demand, x, was drawn from the appropriate distribution in Table 1 and both subjects within a group received the signals θ1 = x + 1 and θ2 = x + 2 . 5.1.4.

Non-Common Knowledge Game With Communication (ms) In these treat-

ments, the information structure was the same as in the ncp treatment, except that after receiving their signal but before making their decision, subjects were asked to send a message to their partner. Messages were of the form, “My estimate is: Z”, where Z was restricted to the interval [x − η, x + η], but did not have to match one’s own signal. There was no cost of sending a message. After both subjects in a group sent a message, they were taken to a new screen where they could make their decision. On this screen, subjects saw their signal as well as the message sent by their match.

6.

Analysis: What is the Role of Private Information?

In this section, we focus on our cp and ncp games to gain insights into the role that private information about the state of demand has on alignment and profits. 6.1.

Basic Results

6.1.1.

Profits. We begin by presenting some basic summary statistics from each of the cp

and ncp sessions that we conducted. These results are on display in Table 2. The final column

16


presents the gap between average profits in each game and the optimal expected profits if subjects played according to the efficient equilibrium for that treatment. Table 2

Summary statistics (a) Distribution of Demand: U (20, 50); π = 5; γ = 2

Treatment cp ncp

Payoff 84.8 78.2

Std. Dev. 6.2 4.2

Min 70.0 71.0

Max 95.7 86.3

Gap (%)† 14.9 17.6

(b) Distribution of Demand: U (100, 400); π = 10; γ = 3

Treatment cp ncp

Payoff 1638.6 1594.9

Std. Dev. 120.4 130.2

Min 1328.1 1331.6

Max 1784.2 1866.9

Gap (%)† 3.7 4.6

(c) Distribution of Demand: U (100, 400); π = 10; γ = 6

Treatment cp ncp

Payoff 804.9 874.1

Std. Dev. 47.5 43.9

Min 747.0 818.1

Max 884.1 927.9

Gap (%)† 14.7 -118.5

†

Calculated as the percentage difference from the either the efficient equilibrium of the cp game or from the monotone equilibrium of the ncp game, depending on the treatment. The optimality gap was obtained via a Monte Carlo simulation consisting of 10,000 trials of 30 or 40 periods, depending on the treatment. Shaded cells indicate a significant difference between the CP and NCP games at the 5% level or better.

There are a number of results that we would like to point out. First, in only one case are payoffs significantly higher in the cp treatment than in the corresponding ncp treatment. Indeed, in our “high-cost” sessions reported in panel (c), subjects actually earned significantly more in the ncp game, despite the fact that the unique equilibrium of the ncp game is the complete coordination failure (hence the negative gap). Thus, the evidence for or against Hypothesis 1 is inconclusive. Second, in two of the cp games, subjects earn approximately 15% less than in the Pareto efficient equilibrium, while in a third subjects come within 3.7%. This suggests that, for some parameter values, subjects are not able to coordinate on the efficient equilibrium. Similarly, in the ncp games payoffs vary with the parameter values and, more strikingly though perhaps unsurprisingly, subjects do not suffer from the complete coordination failure predicted when γ > π/2. 6.1.2.

What is the extent of misalignment? Here we quantify the amount of misalignment

in each group’s choices and try to determine the role of information. Let djt denote the absolute difference between the choices of the subjects in group j in round t, and let d¯ denote the average over all groups and rounds. Table 3 reports these data. Observe that, comparing the degree of misalignment, between cp and ncp treatments, we see that pairs were more misaligned in the ncp treatment than in the cp treatment — a result which

17


Table 3

¯ The extent of misalignment (d)

Parameters

cp 4.67 Demand ∼ U [20, 50]; π = 5; γ = 2 (2.03) 24.43 Demand ∼ U [100, 400]; π = 10; γ = 3 (13.0) 12.85 Demand ∼ U [100, 400]; π = 10; γ = 6 (4.1)

ncp 5.96 (1.03) 28.78 (12.8) 21.15 (4.0)

Standard deviations reported below, in parentheses. Highlighted cells indicate a significant difference (at the 5% level) between the the cp and ncp treatments.

is consistent with our theoretical predictions (cf. Hypothesis 2). Furthermore, we see that this difference is statistically significant in two out of the three situations. It is also interesting to note that subjects are significantly more misaligned in cp(10, 3) than in cp(10, 6) (p = 0.014). Although there is no theoretical reason for this to be the case, it is possible that, behaviourally, there is more scope for misalignment in the former game where K(θ) = θ + 10 for θ ∈ [125, 375], while in the latter game, K(θ) = θ − 5 for θ ∈ [125, 375]. Hence, there are “more” equilibria in cp(10, 3), making it more difficult for players to coordinate on any one of them. 6.2.

Comparing choices with equilibrium predictions

In Figure 2, we plot the results of non-parametric regressions of choice on estimate for our cp and ncp treatments, and we compare the estimates to the equilibrium prediction. The left-hand panel shows the estimated choice functions for the cp games, while the right-hand panel shows the estimated choice functions for the ncp games. In all cases, the dashed line represents the equilibrium capacity choice function and the shaded areas around the choice functions represent the 95% confidence intervals. Leaving aside for the moment the ncp(10, 6) game, we see that the estimated choice functions are close to, but below the theoretical prediction for both the cp and ncp treatments. Furthermore, it appears that the difference between the estimated choice functions and the equilibrium choice functions is increasing in the signal received. This suggests that subjects may be somewhat risk averse since the potential for lost earnings is greater (i.e., there is more room to be undercut) when one observes a higher signal. Interestingly, it seems that this risk aversion is most pronounced in the cp(5, 2) game. In this game, subjects are, on average, choosing a capacity of 89.6% of the efficient capacity. In contrast, in cp(10, 3) subjects choose 96.9% of the efficient capacity, while in cp(10, 6), the number is 94.4%. In both cases, these are significantly greater than the cp(5, 2) game. Comparing cp(10, 3) with cp(10, 6), it appears that subjects adjusted their capacity choice functions too much in response to the increase in the cost of capacity. While in the former game, (except

18


Figure 2

Estimated choice rules vs. equilibrium predictions

Capacity 15 20 25 30 35 40 45 50 55 Estimate CP-LC Pareto CP(5,2); Efficient Choice Received Demand Equilibrium ~ U[20,50]

cp games

ncp games

Capacity Choice 15 20 25 30 35 40 45 50 55 Estimate NCP-LC Equilibrium NCP(5,2); Received of Demand NCP Game ~ U[20,50]

45 Capacity Choice 30 35 40 25 20

20

25

Capacity Choice 30 35 40

45

50

NCP(5,2); Demand ~ U[20,50]

50

CP(5,2); Demand ~ U[20,50]

15

20

25

30 35 40 Estimate Received

CP-LC

45

50

55

20

Pareto Efficient Equilibrium

25

30 35 40 Estimate Received

NCP-LC

400 Efficient 350 300 250 200 150 100 Capacity 75 125 175 225 275 325 375 425 Estimate CP-LC Pareto CP(10,3); Choice Received Demand Equilibrium ~ U[100,400]

45

50

55

Equilibrium of NCP Game

400 350 300 250 200 150 100 Capacity 75 125 175 225 275 325 375 425 Estimate NCP-LC Equilibrium NCP(10,3); Choice Received ofDemand NCP Game ~ U[100,400]

NCP(10,3); Demand ~ U[100,400]

100

100

150

150



350

350

400

CP(10,3); Demand ~ U[100,400]

400

15

75

125

175 CP-LC

225 275 Estimate Received

325

375

425

125


175


NCP-LC

400 Efficient 350 300 250 200 150 100 Capacity 75 125 175 225 275 325 375 425 Estimate CP-HC Pareto CP(10,6); Choice Received Demand Equilibrium ~ U[100,400]

325

375

425

Equilibrium of NCP Game

400 350 300 250 200 150 100 Capacity 75 125 175 225 275 325 375 425 Estimate NCP-HC NCP-LC NCP(10,6); Choice Received Demand ~ U[100,400]

NCP(10,6); Demand ~ U[100,400]

350 150 100

100

150



350

400

CP(10,6); Demand ~ U[100,400]

400

75

75

125

175 CP-HC


325

375


425

75

125

175

225 275 Estimate Received NCP-HC

325

375

425

NCP-LC

for high signals) subjects came quite close to the Pareto efficient equilibrium, in the latter game, the choice function is uniformly below the efficient equilibrium. Note also that “anchoring around mean demand and insufficient adjustment” does not appear to explain the behaviour in these two games. For example, on the interior, it should be that K(θ) − θ = 10 in cp(10, 3), while in cp(10, 6), the difference should be −5. However, the actual differences are, on average, 2.08 and -18.52. That


19

is, while subjects under-adjust when cost is low, they actually over-adjust when cost is high. As we have said, for the ncp(10, 6) game, the theoretical prediction is for the complete coordination failure. As the bottom-right figure indicates, we clearly do not see this. That being said, subjects do respond by lowering their capacity. Somewhat strikingly, the difference between the two games is only 13.26 on average, while in the cp games, the difference was 20.6. Thus subjects appear to be less responsive to cost in the ncp games than in the cp games despite the opposite theoretical prediction. This may be further evidence that the multiplicity of equilibria in the cp games poses real challenges for subjects. 6.3.

Further data analysis

Space does not permit us to provide some data analysis which may be of interest to some readers. In a set of supplemental notes, we provide such further analysis. We briefly highlight the results. First, we estimate our main capacity choice functions parametrically using a piecewise linear function and show that the results are very similar to the results reported in Section 6.2. Second, we investigate whether subjects learn to better-align their capacities and whether profits increase over time. We show that alignment significantly improved in all games except the ncp(10, 6) game, while average profits tended to increase over time, except in ncp(10, 3) and ncp(10, 6). Interestingly, after about round 30, average profits in the ncp(10, 6) game fell off quite sharply. Third, we investigate whether current choices are affected by lagged variables. There we show that there is a positive correlation between current and lagged choice, indicating some inertia in choices. We also show a negative relationship between current choice and the lagged difference between the subject’s own choice and her opponent’s choice, though the effect is only significant in two of six games.

7.

Analysis: Mechanisms to Improve Coordination

We now analyse subject behaviour in our cp-2s and ms treatments. Our goal is to determine whether, as predicted, profits and alignment improve relative to the cp and ncp games. 7.1.

Basic Results

We begin in Table 4 by reporting summary statistics on average earnings. The table also reports the efficiency gap, relative to the efficient equilibrium in cp-2s. It also reports the results of hypothesis tests comparing average profits in each game with the corresponding cp and ncp game. Observe that, in contrast to Hypothesis 1, average profits are actually higher in the ms game than in the cp-2s game, although the difference is not significant. Thus, despite the potential for lying, subjects perform quite well in the ms treatment. We see that average profits are higher in cp-2s than in cp in two out of the three games, while profits are only higher than ncp when

20


Table 4

Summary statistics for the CP-2S and MS Treatments (a) Distribution of Demand: U (20, 50); π = 5; γ = 2

Treatment cp-2s ms

Payoff 90.2 90.8

Std. Dev. 4.1 5.9

Min 77.1 75.7

Max 100.8 99.1

Gap (%) 10.4 9.8

†

Hypothesis Test‡ cp ncp 0.01 0.01 0.01 0.01

(b) Distribution of Demand: U (100, 400); π = 10; γ = 3

Treatment cp-2s ms

Payoff 1613.7 1668.5

Std. Dev. 74.3 107.2

Min 1500.3 1474.2

Max 1723.6 1853.6

Gap (%)† 4.6 1.4


(c) Distribution of Demand: U (100, 400); π = 10; γ = 6

Treatment cp-2s ms

Payoff 886.4 906.0

Std. Dev. 56.3 47.9

Min 807.6 835.2

Max 973.3 983.0

Gap (%) 7.2 5.1

†


†

Calculated as the percentage difference from the efficient equilibrium of the cp-2s game. The optimality gap was obtained via a Monte Carlo simulation consisting of 10,000 trials of 30 or 40 periods, depending on the treatment. ‡ Reports the p-value of the one-sided hypothesis test that average profits in ms or cp-2s are equal to average profits in the cp and ncp treatments.

(π, γ) = (5, 2). In contrast, when (π, γ) ∈ {(5, 2), (10, 6)}, average profits are higher in the ms game than in both of the cp and ncp games, while for (π, γ) = (10, 3), average profits are only higher than in the ncp game. Thus, communication seems to have strong welfare-improving effects. Turn now to alignment. Our results are presented in Table 5. We make two observations. First, in contrast to Hypothesis 2, subjects are never statistically significantly better-aligned in cp-2s than in ms. In fact, for (π, γ) = (10, 6), subjects are significantly better-aligned in the ms game (p = 0.03). Second, while subjects in the ms games are significantly better-aligned than subjects in the cp or ncp games in 5 of 6 cases, the same can be said of cp-2s in only 4 cases. Furthermore, when comparing cp-2s(10, 6) with cp(10, 6), we see that the degree of misalignment is actually worse in the former game. This goes strongly against Hypothesis 2. 7.2.

How Truthful Are Subjects and are There Consequences From Lying?

The results of the previous subsection indicate that (cheap talk) communication appears to be beneficial. Of course, we do not know whether subjects are playing the truthful equilibrium of Proposition 3. We turn our attention to this now. In Table 6, we categorize the messages that were sent. Consistent with our intuition, the plurality of messages were greater than one’s signal, while messages were truthful approximately 25% of the time. Somewhat puzzling is the fact that subjects sent messages that were strictly less than their estimate between 16 and 23% of the time. To the

21


¯ The extent of misalignment (d)

Table 5

Parameters

cp-2s 3.29 Demand ∼ U [20, 50]; π = 5; γ = 2 (1.39) 14.56 Demand ∼ U [100, 400]; π = 10; γ = 3 (5.9) 19.29 Demand ∼ U [100, 400]; π = 10; γ = 6 (7.4)

Test cp vs. Test ncp vs. ms cp-2s ms cp-2s ms 3.27 0.01 0.01 0.01 0.01 (0.77) 19.41 0.02 0.08 0.01 0.01 (8.3) 13.43 0.99 0.63 0.26 0.01 (3.7)

Standard deviations reported below, in parentheses. † Observe that the yellow-shaded cell comparing cp-2s(10, 6) and cp(10, 6) indicates that alignment is actually significantly better in the cp game, contrary to the theoretical prediction.

extent that messages are believed, this can only lead to lower subsequent capacities and profits.9 Table 6

The truthfulness of signals

Parameters Demand ∼ U [20, 50]; π = 5; γ = 2 Demand ∼ U [100, 400]; π = 10; γ = 3 Demand ∼ U [100, 400]; π = 10; γ = 6

θi < Mi 23.03% 16.08% 19.00%

θi = Mi 27.88% 23.19% 20.75%

θi > Mi 49.09% 60.72% 60.25%

In order to get a more visual sense of the relationship between one’s signal and the message that they send, in Figure 3 we show the non-parametric estimate of the message as a function of one’s estimate. As can be seen, subjects inflate their messages more for relatively lower signal realisations and are quite honest for higher signal realisations. Somewhat surprisingly, in the ms(5, 2) game, subjects are consistently deflating their message starting around signals of 45.10 Figure 3

Non-Parametric Estimate of Messages as a Function of One’s Signal Message 15 20 25 30 35 40 50 55 Estimate MS(5,2) 45 Degree Sent Received Line

15

75

20

25

Message Sent 30 35 40

45

50

Message Sent 125 175 225 275 325 375 425

55

Message 75 125 175 225 275 325 375 425 Estimate MS(10,3) MS(10,6) 45 Degree Sent Received Line

75

125

175

MS(10,3)

225 275 Estimate Received MS(10,6)

325

375 45 Degree Line

425

15

20

25

30 35 40 Estimate Received MS(5,2)

45

50

55

45 Degree Line

9

An alternative explanation for this apparently irrational behaviour could be that subjects were trying to use their message to signal their eventual capacity choice. 10

Since messages were bounded such a result may not be surprising. For very high signals, there is not much room to inflate one’s message, while there is more scope for deflating one’s message.

22


We next turn to the question of what affects capacity choices in the ms treatment. Recall that if subjects are playing the truthful equilibrium of Proposition 3, then one’s message and estimate should be perfectly co-linear; however, since subjects lie the majority of the time, this no longer holds. Furthermore, one should give equal weight to their own signal and the message they receive. Also note that to sustain the truthful equilibrium, if a player is revealed to have lied about their signal (i.e., |Mj − θi | > 2η), then a punishment should ensue. Table 7 examines these predictions. Table 7

Random-effects Tobit regressions of capacity choice (ms treatment)

Demand ∼ U [20, 50] Demand ∼ U [100, 400] Demand ∼ U [100, 400] π = 5; γ = 2 π = 10; γ = 3 π = 10; γ = 6 (1) (2) (3) (1) (2) (3) (1) (2) (3) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ estimate 0.739 0.595 0.730 0.716 0.328 0.658 0.566 0.181 0.527∗∗∗ [0.023] [0.030] [0.023] [0.024] [0.034] [0.026] [0.030] [0.044] [0.031] message 0.193∗∗∗ 0.180∗∗∗ 0.202∗∗∗ 0.245∗∗∗ 0.259∗∗∗ 0.302∗∗∗ 0.404∗∗∗ 0.393∗∗∗ 0.442∗∗∗ rec’d [0.022] [0.021] [0.022] [0.023] [0.021] [0.025] [0.029] [0.026] [0.031] ∗∗∗ ∗∗∗ ∗∗∗ 0.178 0.390 0.404 message sent [0.026] [0.027] [0.037] |Mj − θi | -0.930∗ -17.83∗∗∗ -14.37∗∗∗ > 2η [0.534] [3.356] [3.777] cons 0.756 -0.173 0.803 10.40∗∗∗ 1.921 11.50∗∗∗ 0.191 -4.955 0.905 [0.597] [0.603] [0.598] [3.448] [3.075] [3.414] [3.329] [3.123] [3.235] N 660 660 660 914 914 914 400 400 400 LL -1656 -1633 -1654 -4107 -4013 -4094 -1698 -1645 -1691 Standard errors (at subject level) in brackets. ∗∗∗ significant at 1%; ∗∗ significant at 5%; ∗ significant at 10%.

The first column for each of set of parameter values contains the basic regression of capacity choice on one’s own estimate and the message received. Notice that the coefficient on one’s own estimate is always significantly greater than the coefficient on the message received (p < 0.01). Next, if we measure the extent of lying as the absolute percentage difference of the message from one’s own estimate, then subjects lied the most in the ms(5, 2) game and the least in the ms(10, 6) game. It is therefore interesting to note that subjects appear to place the least weight on the message received in the ms(5, 2) game and the most weight on the message received in the ms(10, 6) game. Thus, subjects may be aware of the greater extent of lying in the former game. Table 6 and Figure 3 shows that subjects do not truthfully report their signals. Furthermore, the columns labeled (1) of Table 7 suggest that subjects are aware of this and so they place less weight on the message received than on their own signal. Yet, as we saw in Table 4, the ability to communicate generally leads to higher profits. The question, therefore, is why? We believe that there are at least three reasons for this. First, as seen in Figure 3, messages are, on average, an


23

increasing function of one’s estimate. Therefore, even though subjects are lying, this function can be inverted to (noisily) reveal the true estimate. Second, as the second column of Table 7 for each game indicates, it appears that one’s own choice is positively correlated with the message that they send to their match. This provides subjects with an opportunity to make inferences about their match’s eventual choice. These two reasons suggest that communication may serve as a coordination device that allows subjects to come close to the efficient outcome. In contrast, in the cp-2s treatment, the fact that each player receives the same two signals about demand does not give them the opportunity to signal their intended action. Of course, all would be for naught if subjects lied “too much,” which brings us to our third reason. Namely, it appears that the offequilibrium punishments at the heart of the proof of Proposition 3 are at play. That is, if a subject receives a signal Mj which differs from their signal, θi , by more than 2η, then subject j is known to have lied, which should entail a subsequent punishment by player i. Indeed, as the third column of Table 7 shows, the coefficient on the dummy variable for this event is negative and significant. This would appear to provide some discipline to ensure that people are not “too dishonest”. Finally, we take a deeper look at the impact of sending and receiving (possibly false) messages has on profits and on misalignment. In panel (a) of Table 8 we report the average profits of subjects in the ms treatment conditional on (i) receiving an honest message, (ii) unknowingly receiving a dishonest message (i.e., |θi − Mj | ≤ 2η) and (iii) knowingly receiving a dishonest message (i.e., |θi − Mj | > 2η. As a point of comparison, we also report the average profits from the cp-2s treatment. Panel (b) makes the same comparison for the misalignment of capacity choices. First consider profits. There are two interesting features. First, average profits are higher in the ms treatment when the subject receives an honest message than in the cp-2s treatment. Thus, communication, as suggested by Proposition 3, may serve as a coordination device and allow subjects to reach the efficient equilibrium. In contrast, the continuum of equilibria in the cp-2s treatment may make coordination on any equilibrium, let alone the efficient one, difficult to achieve. Second, with one exception, we see that average profits are decreasing as we go from honest messages to undiscovered dishonest messages to messages that are known to be dishonest. At least one reason for why we see this apparent pattern in average profits can be seen by looking at Panel (b) of Table 8. In all cases, we see that average misalignment is increasing as we go from honest messages to undiscovered dishonest messages to messages that are known to be dishonest. Even more striking is the comparison of alignment between cp-2s(10, 6) and ms(10, 6) when the subject receives an honest message. As can be seen, subjects are highly significantly more

24


Table 8

The consequences of lying: Average profits & Misalignment under various scenarios on the message received (a) Average Profits

cp-2s

ms θj = Mj

Dem. ∼ U [20, 50]; 90.2 = 0.57 π = 5; γ = 2 Dem. ∼ U [100, 400]; 1613.7 = 0.20 π = 10; γ = 3 Dem. ∼ U [100, 400]; 886.4 < 0.04 π = 10; γ = 6

91.6 1691.4 972.4

=

ms θj 6= Mj & |Mj − θi | ≤ 2η 92.1

=

1681.2

0.86

0.82

>

>

ms θj 6= Mj & |Mj − θi | > 2η 73.0

>

1537.6

0.01

0.06

892.5

0.09

=

802.1

2η 5.99

0.01

=

ms θj 6= Mj & |Mj − θi | ≤ 2η 3.11

=

16.97

0.94

0.59

9.29

θk ) , and if sending a deflated message also leads to lower profits, then we would expect a positive coefficient on (Mk − θk )1(Mk θk ) , k = 1, 2, are always negative, indicating that it is not profitable to lie by inflating messages. To some extent, in the game ms(10, 3), this effect is mitigated by the positive and significant coefficient on 1(Mj 6=θj ) . Therefore, at least for this game, small lies may be profitable, but big lies are counterproductive. We also see that the coefficient on (Mk − θk )1(Mk θi ) (Mi − θi )1(Mi θj ) (Mj − θj )1(Mj nπ , an inequality ever more likely to be satisfied as n increases.11 Our experiment set out to test the main predictions of our model in order to highlight the practical role of information structure (i.e., common vs. non-common knowledge) and communication. We found that average profits were not consistently higher in our cp games than in our ncp 11

Indeed, van Huyck, Battalio, and Beil (1990) studies the so-called “minimum effort game”, which is similar to ours (though without demand uncertainty and with common knowledge) and experimentally finds that complete coordination failures are more likely to occur in larger groups.


27

games, and were sometimes worse. This suggests two things. First, it suggests that the continuum of equilibria inherent in the cp games makes coordination on any equilibrium, let alone the efficient one, a challenge (recall also Table 8, where a similar effect was noted comparing the cp-2s and ms games). Second, it may be that subjects in our ncp games did not fully appreciate exactly how different their information could be from their match’s. This may have been why we did not observe the complete coordination failure in the ncp(10, 6) game; however, the collapse in payoffs in the last 10 rounds of this game suggests that they may be learning to appreciate the practical implications of non-common knowledge. Also note that, while average profits were not consistently higher in our cp games, alignment was consistently better than in the ncp games. Our results also showed the beneficial effects of both truthful information sharing and pre-play communication. Indeed, average profits were approximately 7.6% higher in the ms treatment than in the cp and ncp treatments overall, and were 11.6% higher when subjects received a truthful message. Along with the increase in profits, alignment was generally improved in the cp-2s and ms treatments relative to the cp and ncp treatments. Similarly, alignment was best in the ms treatment when subjects received truthful messages from their match. Somewhat surprising was the fact that the ms treatment performed so well when compared against the cp-2s treatment. We believe that this is because the act of communicating — not necessarily truthful — information sets can also be an opportunity to signal possible plans. In other words, when one player sends a message about his or her forecast of demand to the other player, this message has information about his or her plans as well. This is possible partly because, in our settings, large deviations from the truth are discovered and punished (cf. Table 7). This provides for some control mechanism that avoids falling into the inefficient babbling equilibria. There are a number of avenues that we are pursuing to extend our work both experimentally and theoretically. From a theoretical perspective, we hope to generalize our results to the case in which firms are asymmetric and also to more general distributions for both prior demand and forecast accuracy. Our results also showed that, when marginal cost of capacity is too high, the unique equilibrium of the game is that of a complete coordination failure. A sequential moves game has the potential to eliminate such bad equilibria. However, this is by no means guaranteed. Recall the logic of Proposition 3, which says that firms will truthfully report their signals because they expect to coordinate on the efficient equilibrium of the capacity choice subgame. This works because there is two way communication. In contrast, in a sequential game, communication is only one way (from the first-mover to the second-mover); therefore, common knowledge is not restored. What this means is that the first-mover must entertain the possibility that the second-mover will receive a

28


low signal and, therefore, choose a lower capacity than the first-mover would desire. Therefore, the first-mover may have the temptation to choose a higher than expected capacity in order to make the second-mover think that it received a higher signal than it actually did. To be sure, since building capacity is costly, there is a limit to how much the first-mover would increase its capacity, which leaves hope that it is possible to find a monotone equilibrium of the sequential-moves game. Observe that the outcome of such a game will be inefficient for two reasons. First, information transmission is only one-way and, second, to ensure that the first-mover does not deviate, in any equilibrium, the first-mover will be choosing a higher-than-optimal capacity. From an experimental perspective, there is also much more that could be done. Because we wanted to focus on the role of information, we did not look for specific biases in decision making that have been noted in much of the existing literature on experimental newsvendor games, though we noted that some subjects appear to be risk averse. Such an analysis may also be fruitful, especially since our experimental methodology is different from the existing literature. It would also be interesting to play the ms game where the distribution of signals has an infinite support (e.g., normal distribution) because in such cases one can never say for certain that a message obviously misrepresents the other player’s signal. As mentioned above, since messages, while ostensibly about one’s private information, seem to signal players’ intentions, it would therefore be interesting to conduct an experiment in which subjects can communicate both their private information and their intended action. Note that in such an experiment, we may need two stages of communication because one presumably updates their intended action based upon the information they received from the communication of private information. One also wonders whether repeated interactions could improve coordination to the point where the perfect information setting outperforms the setting with pre-play communication. If so, then this would suggest that long term relations would be better at achieving coordination than imperfect information sharing. Finally, do these results extend to other games, even games where the equilibrium set is not well understood? For example, Croson, Donohue, Katok, and Sterman (2004) find that, in the beer game, perfectly sharing demand information does not eliminate the bullwhip effect at all. If our conclusions could be generalized to this setting, then we would predict that a treatment where players in the beer game were able to send messages to their corresponding upstream party with their estimates of demand should perform better than a treatment that simply shares true demand information, even if players sometimes lied about their forecasts. One important insight from this study is the following. Not surprisingly, information sharing leads to better alignment and higher profits. What is surprising is that this is true whether there


29

is truthful information sharing or, more simply, pre-play communication in which firms may lie about their private information. Thus, our results suggest that, instead of sharing actual sales and other information that could be used to come up with a common forecast (e.g., POS data), companies should directly share their own forecasts. Moreover, despite the fact that complaints about “exaggerated” forecasts can be common among practitioners, companies should not be too concerned about people not being perfectly honest in these shared forecasts (within certain limits). What our experiment teaches us is that communication, despite the potential for lying, may be more valuable than information. In our view, this is because communication gives players the opportunity to implicitly signal their intended course of action, which has value above and beyond any information about demand contained in the message. Moreover, communication is a much less costly and burdensome institution to introduce than one which ensure truthful information sharing. Of course, further testing of these results, along the lines described above, would allow us to refine our prescriptions. Acknowledgments Financial support for this project has been provided by SMU and the National Science Foundation (SES1025044), as well as ZLC and HBS. We would also like to thank Elizabeth Pickett, Drew Goin, Ravi Hanumara and Robert Reeves for their help running the experiments. We also thank participants at the 4th annual Behavioral Operations Conference and the 2010 POMS Annual Meeting, as well as seminar participants at the University of Texas at Austin and MIT for their valuable comments. Finally, we are grateful to Rachel ¨ ¨ Croson, Karen Donohue, Doroth´ee Honhon, Ozalp Ozer and Sridhar Seshadri for their valuable comments.

Appendix. Omitted Proofs Proof of Proposition 0.

Consider first the sales firm. Ignoring for a moment the capacity choice of

manufacturing, the derivative of sales’ expected profit function is given by: πs (1 − F (Ks |θ)) − γ. Notice that if θ < η(1 − 2s∗ ), then the derivative is strictly negative for all Ks ≥ 0. Therefore, for this range of θ, there is a unique best response in which Ks (θ) = 0. On the other hand, for θ > η(1 − 2s∗ ), the solution to the first order condition is easily seen to be Ks∗ (θ) = F −1 (s∗ |θ). Of course, if Km < Ks∗ (θ), then sales prefers to choose capacity Km . Hence, the best-response function for sales is Ks∗ (Km ) = min{Km , F −1 (s∗ |θ)}. A similar calculation holds for the manufacturing firm. Thus the result follows.

30


Proof of Proposition 1.

We focus on the characterisation of the monotone equilibrium. The other results

in the proposition are easy to prove and are already discussed in the text. Consider the problem of the sales firm. Denote manufacturing’s capacity choice rule by Km (θm ) and assume that it is strictly increasing. Fix the signal of sales at θs and suppose that it contemplates a capacity of k. We know that there exists a critical value of the signal received by manufacturing, θ(k), such that if θm > θ(k), −1 then Km (θm ) > k. Since manufacturing’s capacity rule is strictly increasing, we know that θ(k) = Km (k). −1 Therefore, sales’ capacity choice k will determine total capacity whenever x ≥ k and θm ≥ Km (k).

It can be shown that the derivative of the profit function for sales, given signal θs and contemplating a capacity choice of k ∈ [0, θs + η], is given by: Z x+η Z θs +η ¯s ∂Π π dydx − γ. = 2 −1 ∂k 4η max{θs −η,k} max{x−η,Km (k)}

(1)

We may now impose the symmetric equilibrium condition Km (θ) = Ks (θ) ≡ K(θ). This implies that we may −1 (k) with θs in the lower limit of the inner integral. Furthermore, it is apparent that θs ≥ x − η replace Km

for all x ∈ [θs − η, θs + η]. Therefore, the lower limit of the inner integral must, in fact, be θs . We first show that there exists θ such that for θs ≤ θ, Ks (θs ) = 0. Indeed, θ is the solution to: Z θ+η Z x+η π dydx − γ = 0, 4η 2 0 θ which, upon solving, yields: r θ = η − 2η

1−

2γ . π

We now solve for the equilibrium capacity choice functions for sales and manufacturing for θ > θ. This amounts to setting (1) equal to zero and solving for k. Upon doing so, we find that: r 2γ K(θ) = θ − η + 2η 1 − . π Observe, however, that if γ > π2 , then the term inside the square root will be negative. In fact, this gives the condition under which a symmetric equilibrium in monotone strategies can be said to exist. To see this more clearly, observe that (1) evaluates to: ¯s ∂Π π = −γ + 2 k 2 − 3η 2 + 2k(η − θ) − 2ηθ + θ2 . ∂k 8η Upon noting that the derivative of (1) with respect to k over the interval [θ − η, θ + η] is negative (i.e., the profit function is concave) and evaluating the above expression at k = θ − η, we see that: ¯s 1 ∂Π = (π − 2γ), ∂k 2 which is negative for γ > π2 . Therefore, if γ > π2 , the firms will lower their capacities to at least θ − η. In fact, they will set K(θ) = 0. ¯ s /∂k = 1 (π − 2γ) < 0. To see this, observe that if k < θ − η, then (1) also simplifies to ∂ Π 2


Proof of Proposition 3.

31

Let Mi denote firm i’s message. Define the beliefs of firm j, upon receiving

message Mi as follows: θˆi =

Mi , if |θj − Mi | ≤ 2η , θ, o.w.

where θ is sufficiently small such that the optimal capacity is to choose 0. First, suppose that firm i has sent the message Mi = θi and received a message Mj such that |θj − Mi | ≤ 2η. Given i’s beliefs about the message from firm j and also the capacity choice rule by j, clearly it is optimal for firm i to choose Ki = F −1 (s∗ |θm , θs ). This follows because this capacity corresponds to the Pareto efficient equilibrium of the cp-2s game. Next suppose that firm i has sent the message Mi = θi and received a message Mj such that |θj − Mi | > 2η. Given the beliefs of firm i, it is clearly optimal to choose a capacity of 0. Now suppose that firm i reports Mi > θi . Then, one of two things will happen. First, it may be that |Mi − θj | ≤ 2η, in which case firm j will choose Kj > F −1 (s∗ |θm , θs ); however, firm i will still find it optimal to choose Ki = F −1 (s∗ |θm , θs ). Second, it may be that |Mi − θj | > 2η, in which Kj = 0. Firm i, upon receiving firm j’s message will be able to deduce that |Mi − θj | > 2η, and so will also choose Ki = 0, which means that firm i will earn a payoff of 0. Therefore, it is better for firm i to report Mi = θi . Finally, suppose that firm i reports Mi < θi . Then, with probability 1, player j will choose capacity strictly less that F −1 (s∗ |θ` , θh ), in which case firm i will earn profits strictly less than the Pareto optimal equilibrium payoff. Therefore, it is better for firm i to report Mi = θi .

References Aviv, Y. (2001): “The effect of collaborative forecasting on supply chain performance,” Management Science, 47(10), 1326–1343. Bendoly, E., K. Donohue, and K. L. Schultz (2006): “Behavior in operations management: Assessing recent findings and revisiting old assumptions,” Journal of Operations Management, 24, 737–752. Brandts, J., and D. Cooper (2006): “A Change Would Do You Good: An Experimental Study of How to Overcome Coordination Failure in Organizations,” American Economic Review, 96, 669–693. Cachon, G. P. (2003): “Supply chain coordnation with contracts,” in Handbooks in Operations Research and Management Science: Supply Chain Management, ed. by S. Graves, and T. de Kok. North-Holland, Amsterdam. Cachon, G. P., and M. Fisher (2000): “Supply chain inventory management and the value of shared information,” Management Science, 46(8), 1032–1048. Cachon, G. P., and M. Lariviere (2001): “Contracting to assure supply: How to share demand forecasts in a supply chain,” Management Science, 47(5), 629–646. Carlsson, H., and E. van Damme (1993): “Global Games and Equilibrium Selection,” Econometrica, 61(5), 989–1018. Celikbas, M., J. G. Shanthikumar, and J. M. Swaminathan (1999): “Coordinating production quantities and demand forecasts through penalty schemes,” IEE Transactions, 31 9(851-864).

32


Chao, X., S. Seshadri, and M. Pinedo (2008): “Optimal Capacity in a Coordinated Supply Chain,” Naval Research Logistics, 55, 130–141. Chen, F. (2005): “Salesforce incentives, market information and production/inventory planning,” Management Science, 51(1), 60–75. Croson, R. T., K. Donohue, E. Katok, and J. Sterman (2004): “Order Stability in Supply Chains: Coordination Risk and the Role of Coordination Stock,” MIT Sloan Working Paper No. 4513-04. Fawcett, S. E., and G. M. Magnan (2002): “The rhetoric and reality of supply chain integration,” International Journal of Physical Distribution, 32(5), 339–361. Fischbacher, U. (2007): “z-Tree: Zurich toolbox for ready-made economic experiments,” Experimental Economics, 10(2), 171–178. Gonik, J. (1978): “Tie salesmen’s bonuses to their forecasts,” Harvard Business Review, 56(3), 116–123. Hartigan, J. (1983): Bayes Theory. Spinger Verlag, New York. Hyndman, K., A. Terracol, and J. Vaksmann (2009): “Learning and Sophistication in Coordination Games,” Experimental Economics, 12(4), 450–472. Jain, A., M. Sohoni, and S. Seshadri (2009): “Differential Pricing for Information Sharing Under Competition,” Unpublished. Kahn, K. B., and J. T. Mentzer (1996): “Logistics and interdepartmental integration,” International Journal of Physical Distribution, 26(8), 6–14. Kraiselburd, S., and N. Watson (2010): “Alignment in Cross-Functional and Cross-Firm Supply Chain Planning,” Zaragoza Logistics Center, Working Paper. Kurtulus, M., and B. Toktay (2007): “Investing in forecast collaboration,” Working Paper, Vanderbilt University. Lal, R., and R. Staelin (1986): “Salesforce compenation plans in environments with asymmetric information,” Marketing Science, 5(3), 179–198. Lariviere, M. (2002): “Inducing forecast revelation through restricted returns,” Working Paper, Northwestern University. Lee, H. L., and S. Whang (1999): “Decentralized multi-echelon supply chains: Incentives and information,” Management Science, 45, 633–640. Li, L. (2002): “Information sharing in a supply chain with horizontal competition,” Management Science, 48(9), 1196–1212. Li, L., and H. Zhang (2008): “Confidentiality and Information Sharing in Supply Chain Coordination,” Management Science, 54, 1759–1773. Li, Q., and D. Atkins (2002): “Coordinating replenishment and pricing in a firm,” Manufacturing & Service Operations Management, 4(4), 241–257.


33

Miyaoka, J. (2003): “Implementing collaborative forecasting through buyback contracts,” Working Paper, Stanford University. Oliva, R., and N. Watson (Forthcoming): “Cross-Functional Alignment in Supply Chain Planning: A Case Study of Sales and Operations Planning,” Journal of Operations Management. ¨ Ozer, O., and W. Wei (2006): “Strategic commitment for optimal capacity decision under asymmetric forecast information,” Management Science, 52(8), 1238–1257. ¨ Ozer, O., Y. Zheng, and K.-Y. Chen (2008): “Trust in Forecast Information Sharing,” Working Paper. Porteus, E. L., and S. Whang (1991): “On manufacturing marketing incentives,” Management Science, 37(9), 1166–1181. Schweitzer, M. E., and G. P. Cachon (2000): “Decision Bias in the Newsvendor Problem with a Known Demand Distribution: Experimental Evidence,” Management Science, 46(3), 404–420. Shapiro, B. P. (1977): “Can marketing and manufacturing co-exist?,” Harvard Business Review, 55(5), 104–114. Tomlin, B. (2003): “Capacity Investments in Supply Chains: Sharing the Gain Rather Than Sharing the Pain,” Manufacturing & Service Operations Management, 5(4), 317–333. van Huyck, J., R. Battalio, and R. Beil (1990): “Tacit Coordination Games, Strategic Uncertainty and Coordination Failure,” American Economic Review, 80(1), 234–248.

Coordination in Supply Chains When Demand Forecasts Are Not Common Knowledge: Theory and Experiment Supplemental Material Kyle Hyndman Southern Methodist University, [email protected], http://faculty.smu.edu/hyndman

Santiago Kraiselburd Zaragoza Logistics Center & INCAE Business School, [email protected]

Noel Watson Zaragoza Logistics Center, [email protected]

1.

Example: Truthful but Inefficient Equilibria May Not Exist

Suppose that there is a truthful equilibrium in which firms’ capacity choices are given by: 1 Ki (θ` , θh ) = [s∗ (θ` + η) + (1 − s∗ )(θh − η)]. 2 That is, they only choose half of the efficient equilibrium quantities. To further simplify matters, let π = 10, γ = 3 and η = 5. Next suppose that θ1 = 20, but that M1 = 20.1. That is, firm 1 inflates its signal by 0.1. In order to see whether this represents a profitable deviation, we must calculate the expected profits of firm 1, taking into consideration the capacities that will be chosen in the next stage. It is straightforward to see that the capacity of each firm in the next stage will be: 0, if |θ2 − 20.1| > 10 K= 1 7 . 3 [ (min{20.1, θ } + 5) + (max{20.1, θ } − 5)], o.w. 2 2 2 10 10 This is because firm 1, upon receiving firm 2’s message, understands if it has been found to be a liar before it actually has to choose a capacity. In this case, since it knows it will be punished, firm 1 also chooses a capacity of 0; otherwise, it chooses the same capacity as firm 2. Taking expectations over the set of states and the possible signals of firm 2, the expected profits of firm 1 from inflating its signal by 0.1 are: 74.8352. On the other hand, the expected profits from faithfully reporting its signal are: 74.6643. Therefore, firm 1’s deviation is profitable, and so the equilibrium in this example cannot be truthful. 1

2


2.

Supplemental Data Analysis

2.1.

The CP and NCP Treatments

2.1.1.

Parametric estimation of capacity choice functions In the main text we estimated

the capacity choice functions non-parametrically. This was done because of the non-linear nature of the equilibrium choice functions. However, to a first approximation, one can use a piecewise linear structure to obtain parametric estimated. Specifically, we estimate the following equation via a random-effects Tobit procedure: choiceit = α + β1 θit + β2 (θit − (x + η)) · [θit < x + η] + β3 (θit − (x − η)) · [θit > (x − η)] + µi + νit , where [A] is an indicator variable which takes value 1 if A is true. The results are on display in Table 1. As can be seen, the coefficient on θ is positive and highly significant, though always less than one. Furthermore, there is some statistical evidence in favour of a kink at θ = x + η and θ = x − η, though the magnitude and significance is not consistent across treatments or parameter values. Figures comparing the parametric and non-parametric capacity choice functions (available upon request), show that the parametric functional form is generally very close to our non-parametric estimates. Table 1

Random-effects Tobit regressions of choice on estimate

Demand ∼ U [20, 50] π = 5; γ = 2 cp ncp θ 0.837∗∗∗ 0.882∗∗∗ [0.0321] [0.0333] [θ < x + η](θ − (x + η)) -0.0898 -0.158 [0.163] [0.174] [θ > x − η](θ − (x − η)) -0.116 -0.470∗∗∗ [0.153] [0.176] cons 2.486∗ 0.438 [1.312] [1.342] N 600 540 LL -1646 -1469

Demand ∼ U [100, 400] π = 10; γ = 3 cp ncp 0.961∗∗∗ 0.968∗∗∗ [0.0236] [0.0125] 0.16 -0.337∗ [0.347] [0.190] 0.00589 -0.167 [0.307] [0.175] 11.27∗ 8.797∗∗ [6.688] [4.017] 480 960 -2289 -4414

Demand ∼ U [100, 400] π = 10; γ = 6 cp ncp 0.978∗∗∗ 0.989∗∗∗ [0.0100] [0.0131] -0.777∗∗∗ -0.400∗ [0.144] [0.242] 0.118 -0.233 [0.142] [0.187] -13.64∗∗∗ -9.314∗∗ [3.375] [4.321] 400 320 -1543 -1320

Standard errors (at subject level) in brackets. ∗∗∗ significant at 1%; ∗∗ significant at 5%; ∗ significant at 10%.

2.1.2.

Learning In this section, we discuss whether subjects are able to learn. Learning can take two,

potentially different, forms. First, players can learn to align their actions better with their match. However, they may learn to align by consistently choosing low capacities regardless of the state of demand. Therefore, we also study whether subjects’ profits are increasing over time.

Do subjects learn to align their choices? Recall that subjects played the game for 30 or 40 periods with random rematching in each round. In Table 2 we show the results of a series of random-effects regressions where we regress djt on the round and other control variables. If subjects learn to align choices, then we would expect a negative estimated coefficient. Indeed, this is generally what we see. Except for the ncp(10, 6) game, the coefficient is negative, and is significant at the 1% level in four of five of these games, and at the

3


10% level in the fifth game. Next, note that learning appears to be stronger in the cp games than in the ncp games. If we pool the data across the cp and ncp treatments for each of the three parameter values, the coefficient on round is significantly smaller (meaning faster learning) in the cp game than the ncp game for (π, γ) ∈ {(10, 3), (10, 6)}.

Table 2

Random-effects regressions of djt on round

Demand ∼ U [20, 50] π = 5; γ = 2 cp ncp round -0.159∗∗∗ -0.136∗∗∗ [0.0217] [0.0234] ∗∗∗ demand 0.0840 0.181∗∗∗ [0.0195] [0.0250] cons 4.168∗∗∗ 1.753∗∗ [0.857] [0.851] N 600 540 R2 0.0898 0.144

Demand ∼ U [100, 400] π = 10; γ = 3 cp ncp -1.291∗∗∗ -0.283∗ [0.285] [0.146] -0.0178 -0.0192∗∗ [0.0243] [0.00836] 55.44∗∗∗ 39.42∗∗∗ [12.28] [5.320] 480 960 0.127 0.0133

Demand ∼ U [100, 400] π = 10; γ = 6 cp ncp -0.228∗∗∗ 0.107 [0.0727] [0.127] 0.00206 -0.00532 [0.0104] [0.0128] 17.04∗∗∗ 20.33∗∗∗ [3.557] [3.380] 400 320 0.0291 0.00316

∗∗∗ significant at 1%; ∗∗ significant at 5%; ∗ significant at 10% Standard errors in brackets, clustered at subject level.

Finally, observe that in cp/ncp(5, 2), it appears that the degree of misalignment increases as the state increases, which is consistent with the wider confidence intervals for higher signals seen in Figure 2 in the main text. This could be due to heterogenous risk preferences among subjects in these treatments, though it is somewhat surprising that we do not see the same effect in the other games.

Do subjects earn more in later rounds? The previous subsection shows a tendency towards improved coordination in capacity choices as the experiment proceeds. We now analyze whether subject’s earnings increased as the experiment proceeds. We regress profits on round number, the (unknown) state of demand and also on the match’s choice. The results are reported in Table 3. We focus our attention on the coefficient on round; the interpretation of the other coefficients is straightforward and not central to our discussion of learning. As can be seen, the coefficient on round is, with the exception of ncp(10, 6), positive in all games. For all cp games, the coefficient is significant at the 1% level, while for the ncp games, the coefficient is significant only for ncp(5, 2). One interesting result that is not apparent in Table 3 is the strong collapse in profits starting around period 30 in the ncp(10, 6) game. Recall that in this game, the unique equilibrium is the complete coordination failure. Figure 1 suggests that this may be the eventual outcome if subjects played the game long enough. Indeed, if we regress capacity choice on one’s signal and interact the round with the signal, then the coefficient on this interaction term is negative. This is further indication that subjects may be converging to the unique equilibrium in the ncp(10, 6) game. One can also see that for cp(10, 3), there appears to be a regime shift (from low to high profits) between periods 15 and 20, but otherwise not much learning.

4


Table 3

Random-effects regressions of profits on round

Demand ∼ U [20, 50] π = 5; γ = 2 cp ncp round 0.444∗∗∗ 0.381∗∗∗ [0.0630] [0.0579] ∗∗∗ demand 0.533 -0.229∗∗ [0.140] [0.107] m.c.† 2.110∗∗∗ 2.736∗∗∗ [0.179] [0.177] ∗∗∗ cons -9.315 -5.065∗ [1.669] [2.937] N 600 540 2 R 0.772 0.74

Demand ∼ U [100, 400] π = 10; γ = 3 cp ncp 6.225∗∗∗ 1.101 [1.351] [0.693] ∗∗∗ 4.041 3.443∗∗∗ [0.601] [0.395] 2.770∗∗∗ 3.473∗∗∗ [0.586] [0.397] ∗∗∗ -235.9 -173.1∗∗∗ [41.32] [20.53] 480 960 0.916 0.934

Demand ∼ U [100, 400] π = 10; γ = 6 cp ncp 1.605∗∗∗ -0.516 [0.363] [0.611] 0.117 1.000 [0.539] [0.851] 3.886∗∗∗ 2.989∗∗∗ [0.539] [0.861] ∗∗∗ -104.6 -107.2∗∗∗ [19.53] [19.98] 400 320 0.943 0.886

∗∗∗

significant at 1%; ∗∗ significant at 5%; ∗ significant at 10%. Clustered standard errors (at subject level) in brackets. † m.c. denotes one’s match’s choice.

Figure 1

Non-Parametric Estimates Profits as a Function of Round

2000 1800 1600 1400 1200 Average 0 1 5 10 15 20 25 30 35 40 Round CP NCP Shaded CP/NCP(10,3); 0 area Profit represents Demand 90% confidence ~ U[100,400] interval

1000 900 800 700 600 Average 0 1 5 10 15 20 25 30 35 40 Round CP NCP Shaded CP/NCP(10,6); 0 area Profit represents Demand 90% confidence ~ U[100,400] interval

CP/NCP(10,6); Demand ~ U[100,400]

Average Profit 800 900 700 600

1200

Average Profit 1400 1600 1800

1000

2000

CP/NCP(10,3); Demand ~ U[100,400]

0

5

10

15

20 Round CP

Shaded area represents 90% confidence interval

2.1.3.

25 NCP

30

35

40

0

5

10

15

20 Round CP

25

30

35

40

NCP

Shaded area represents 90% confidence interval

Autocorrelation in choices To examine whether subjects use history-dependent strate-

gies, we estimate the capacity choice function as a function of the current signal and other lagged variables. We include the lagged choice and lagged demand as well as the lagged difference between a subject’s choice and her opponent’s choice. We have no a priori prediction about the relationship between current choice and lagged choice. On the other hand, because the state variable was bounded we might expect a negative correlation between current choice and lagged demand (given high demand in t − 1, it is more likely that demand in t will be lower). Concerning the lagged difference between own choice and opponent’s choice, we do not have a clear prediction. On one hand, because subjects were randomly matched each period, the previous choice by one’s opponent need not be informative about the current choice, because it is very likely that the subject is facing a new opponent. On the other hand, one might expect a negative relationship. If c 6= m.c. then it is very likely that the subject suffered from lost earnings, either because she was undercut by her opponent or because she lost out on potential earnings by choosing too conservatively. In the former

5


case, this negative feedback causes subjects to lower their capacity choice, while in the latter case she chooses a higher capacity, all else equal, in the next period — hence the negative relationship. The results of this exercise are on display in Table 4. Table 4

Random-effects Tobit regressions of choice on estimate and lagged variables

Demand ∼ U [20, 50] π = 5; γ = 2 cp ncp θ 0.821∗∗∗ 0.841∗∗∗ [0.0199] [0.0213] lagged choice 0.118∗∗ 0.111∗∗ [0.0485] [0.0547] lagged demand -0.0662 -0.0649 [0.0425] [0.0448] lagged c − m.c. -0.0704∗∗ -0.131∗∗∗ [0.0345] [0.0345] cons 1.583 0.648 [1.209] [1.278] N 580 522 LL -1568 -1387

Demand ∼ U [100, 400] π = 10; γ = 3 cp ncp 0.972∗∗∗ 0.958∗∗∗ [0.0190] [0.0103] 0.187∗∗∗ 0.0149 [0.0586] [0.0417] -0.177∗∗∗ -0.0226 [0.0576] [0.0406] -0.0179 -0.0248 [0.0426] [0.0289] 6.017 12.92∗∗∗ [7.219] [4.479] 468 936 -2213 -4279

Demand ∼ U [100, 400] π = 10; γ = 6 cp ncp 0.964∗∗∗ 0.980*** [0.00883] [0.0110] 0.133∗∗∗ 0.0619 [0.0432] [0.0658] -0.139∗∗∗ -0.0794 [0.0410] [0.0644] -0.0463 -0.0304 [0.0437] [0.0432] -5.627 -1.864 [3.784] [4.839] 390 312 -1505 -1275

Standard errors (at subject level) in brackets. ∗∗∗ significant at 1%; ∗∗ significant at 5%; ∗ significant at 10%. The variable lagged c − m.c. denotes the lagged difference between the subjects choice and his/her match’s choice.

As can be seen, for all games there is a positive relationship between the current and previous capacity choice, and the effect is significant in four of the six games. Moreover, when significant, the effect seems to be fairly large in magnitude at more than 10% of the effect on one’s signal. This positive correlation indicates that subjects are prone to some inertia in their decision making. For lagged demand, we only see a significant correlation in two of our cp games, and it is negative as was expected. Similarly, in only two games (this time in both games for which demand was distributed ∼ U [20, 50]) do we see a significant relationship between the lagged difference between own and opponent’s choice. Here too the coefficients were negative, which is indicative of some form of adaptive learning.

2.2.

The CP-2S and MS Treatments

2.2.1.

Comparing choices with equilibrium predictions. We would like to replicate Figure

2 from the main text for our cp-2s and ms treatments. Unfortunately, this is not possible since subjects have at least two pieces of information on which to condition their capacity choice in these treatments. However, we are able to compare the actual capacity choice to the theoretical capacity choice given the signals of both players. For this exercise, which is reported in Figure 2, we use as our theoretical benchmark the efficient equilibrium of the cp-2s games. As can be seen, subjects appear to be very close to the efficient equilibrium in the cp-2s(10, 6) and ms(10, 6) games, while in the other games, capacity choices appear to be significantly less than the theoretical prediction, with the largest deviation apparently for the ms(5, 2) game. All of these results are consistent with the patterns reported for the cp and ncp treatments in the main body of the paper.

6


Figure 2

Estimated choice rules vs. equilibrium predictions

cp-2s games

Actual 20 25 30 35 40 50 Theoretical CP-2S-LC 45 CP-2S(5,2); Degree Capacity Line Capacity Demand Choice Choice ~ U[20,50]

Actual 20 25 30 35 40 50 Theoretical MS-LC 45 MS(5,2); Degree Capacity Demand Line Capacity Choice ~Choice U[20,50]

Actual Capacity Choice 30 35 40 45 25 20

20

25

Actual Capacity Choice 30 35 40 45

50

MS(5,2); Demand ~ U[20,50]

50

CP-2S(5,2); Demand ~ U[20,50]

ms games

20

25

30 35 40 Theoretical Capacity Choice CP-2S-LC

45

50

25

45 Degree Line

30 35 40 Theoretical Capacity Choice MS-LC

Actual 100 150 200 250 300 350 400 Theoretical CP-2S-LC 45 CP-2S(10,3); Degree Capacity Line Capacity Demand Choice Choice ~ U[100,400]

45

50

350

400

350

400

45 Degree Line

Actual 100 150 200 250 300 350 400 Theoretical MS-LC 45 MS(10,3); Degree Capacity Line Capacity Demand Choice Choice ~ U[100,400]

MS(10,3); Demand ~ U[100,400]


100

Actual Capacity Choice 150 200 250 300 350

400

CP-2S(10,3); Demand ~ U[100,400]

400

20

100

150

200 250 300 Theoretical Capacity Choice CP-2S-LC

350

400

150

45 Degree Line

200 250 300 Theoretical Capacity Choice MS-LC

Actual 100 150 200 250 300 350 400 Theoretical CP-2S-HC 45 CP-2S(10,6); Degree Capacity Line Capacity Demand Choice Choice ~ U[100,400]

45 Degree Line

Actual 100 150 200 250 300 350 400 Theoretical MS-HC 45 MS(10,6); Degree Capacity Line Capacity Demand Choice Choice ~ U[100,400]

MS(10,6); Demand ~ U[100,400]


100

Actual Capacity Choice 150 200 250 300 350

400

CP-2S(10,6); Demand ~ U[100,400]

400

100

100

150

200 250 300 Theoretical Capacity Choice CP-2S-HC

2.2.2.

350

45 Degree Line

400

100

150

200 250 300 Theoretical Capacity Choice MS-HC

45 Degree Line

Do subjects learn to align their choices? In Table 5 we replicate Table 2, which looks

at the question of whether subjects become better-aligned as the experiment progressed. As can be seen, in all games we find a negative and significant coefficient on round, which indicates that alignment is improving over time. Consistent with the results from the main text, the coefficients on round appear to be smaller in

7


magnitude for the (π, γ) = (10, 6) games than for the (π, γ) = (10, 3) games. Random-effects regressions of djt on round

Table 5

Demand ∼ U [20, 50] π = 5; γ = 2 cp-2s ms ∗ round -0.0496 -0.108∗∗∗ [0.0265] [0.0197] demand 0.100∗∗∗ -0.0119 [0.0285] [0.0115] cons 0.568 5.359∗∗∗ [0.703] [0.644] N 660 660 2 R 0.077 0.0777

Demand ∼ U [100, 400] π = 10; γ = 3 cp-2s ms ∗∗ -0.455 -0.412∗∗∗ [0.177] [0.0968] -0.01 -0.029 [0.0115] [0.0211] 26.37∗∗∗ 34.87∗∗∗ [7.655] [7.823] 400 914 0.0399 0.0284

Demand ∼ U [100, 400] π = 10; γ = 6 cp-2s ms ∗∗ -0.259 -0.239∗∗ [0.104] [0.111] 0.0146 0.0216∗ [0.0229] [0.0116] ∗∗∗ 20.87 12.85∗∗∗ [7.949] [1.728] 480 400 0.0147 0.0456

∗∗∗ significant at 1%; ∗∗ significant at 5%; ∗ significant at 10% Standard errors in brackets, clustered at subject level.

2.2.3.

Do subjects earn more in later rounds? Finally, Table 6 replicates Table 3 and exam-

ines whether profits are improving as the experiment progressed in the cp-2s and ms treatments. In partial contrast to the results on the cp and ncp treatments, we find a positive learning effect in the ms(10, 3) game (where no learning was present in ncp(10, 3)) and we fail to find a learning effect in the cp-2s(10, 6) game (where a positive learning effect was found in cp(10, 6)). Table 6

Random-effects regressions of profits on round

Demand ∼ U [20, 50] π = 5; γ = 2 cp-2s ms ∗∗ round 0.134 0.256∗∗∗ [0.0681] [0.0538] demand -0.0852 0.615∗∗∗ [0.261] [0.162] m.c.† 2.917∗∗∗ 2.405∗∗∗ [0.281] [0.175] ∗∗∗ cons -6.036 -14.87∗∗∗ [2.124] [1.502] N 660 660 R2 0.865 0.902 ∗∗∗

Demand ∼ U [100, 400] π = 10; γ = 3 cp-2s ms ∗∗∗ 2.671 1.737∗∗∗ [0.941] [0.391] 2.714 4.848∗∗∗ [1.854] [0.509] 4.269∗∗ 2.132∗∗∗ [1.882] [0.516] ∗∗∗ -149.8 -150.6∗∗∗ [35.99] [27.13] 400 914 0.951 0.958

significant at 1%; ∗∗ significant at 5%; ∗ significant at 10%. Clustered standard errors (at subject level) in brackets. † m.c. denotes one’s match’s choice.

Demand ∼ U [100, 400] π = 10; γ = 6 cp-2s ms 0.898 -0.125 [0.704] [0.319] 0.857 -0.977∗ [1.035] [0.556] 3.062∗∗∗ 4.955∗∗∗ [1.061] [0.548] ∗∗∗ -117.8 -78.18∗∗∗ [34.78] [19.64] 480 400 0.834 0.944

General Instructions This is an experiment on the economics of decision-making. Your earnings will depend partly on your decisions, partly on the decisions of others and partly on chance. By following the instructions and making careful decisions you will earn varying amounts of money, which will be paid at the end of the experiment. Details of how you will make decisions and earn money are explained below. In this experiment, you will participate in 30 independent decision problems (rounds). In all rounds, you will be randomly matched with another participant. In what follows, we will refer to the person with whom you are matched as your match. After each round, you will be randomly matched with another participant for the next decision problem, and so on. At no point in the experiment will you know the identity of your matches.

Decision Problem In each round you will be asked to choose a number between 20 and 50. Your match will face the same choice problem. If we denote your choice by K and your match’s choice by M , then your decisions will result in the following earnings (the explanation of Q will be given later): earnings = 5 · min{K, M, Q} − 2 · K That is, when determining your profits, we first look for the smallest number of K (your decision), M (your match’s decision) and Q. Whatever that smallest number is, we will multiply it by 5. From that number, we will deduct 2 times the number you chose. For example, if you chose K = 30, your match chose M = 25 and Q = 27, then the smallest number is M = 25, which means that you will earn 5 · 25 − 2 · 30 = 125 − 60 = 65 points. On you other hand, your match would earn 5 · 25 − 2 · 25 = 125 − 50 = 75 points. The software contains a calculator feature to help you calculate your earnings under any hypothetical scenario of your choosing.

What is Q? Q is a number (up-to two decimals) between 20 and 50 randomly determined by the computer. That means any number between 20 and 50 is equally likely to be picked by the computer. The computer picks Q before each round and the numbers are independent across rounds. That is, the Q chosen by the computer in a round has nothing to do with the Q picked in any other round. Before you make a decision you will not be told what Q is but instead you will receive an estimate of Q, which we will denote by E. Let’s be more precise. After the computer randomly determines Q, it also picks a random number (up-to two decimals) between Q − 5 and Q + 5. This is your estimate E. Any number between Q − 5 and Q + 5 is equally likely to be picked by the computer. Although E does not tell you what Q exactly is, it gives an estimate of it. For example 1

if you receive an estimate E = 37.07, then you know that Q is not less than 37.07 − 5 = 32.07 and it is not more than 37.07 + 5 = 42.07. Note that although Q will be the same for both you and your match, your estimates can be different. That is, for the same Q, the computer also randomly picks another estimate exactly in the same manner for your match. Your estimate and your match’s estimate are chosen independently. Therefore, it is very likely that they will be different numbers; however, both estimates will be between Q − 5 and Q + 5. Note also that while Q will be between 20 and 50, your estimate of Q can be between 15 and 55.

The Computer Screen In each round, you will see the following computer screen:

 

On the left side of the screen is where you will see your estimate and make your final decision. In this example, you see that your estimate of Q is 39.01. On the right hand side of the screen, you can enter hypothetical values for your decision, your match’s decision and the value of Q in order to see what your earnings would be under each hypothetical scenario. You can compute your hypothetical earnings under as many different scenarios as you choose.

Your Decision After you are given your estimate, E, you are ready to make a decision. Your decision is simply to choose a number between 20 and 50. The payoffs for this decision are as described above. Note

2

that you and your match make your decisions at the same time; that is, you must choose your number without knowing the choice made by your match.

Payoffs Your potential earnings in each round depend on your choice, on your match’s choice, and on Q. After both you and your match have made your choices, you will see the following screen. You see your estimate of Q, the true value of Q, and your profit. You also see the decision made by both you and your match. In this example, you see that while your estimate of Q was 39.01, its true value was 43.86.

 

At the end of the 30 rounds, we will add all your earnings in order to determine your total points. This total will be converted to a dollar amount according to the rule: $1 = 165 points. This amount will then be added to the $5.00 participation fee to give your final payment. Payments will be made in private, in cash, after the completion of the experiment.

Rules Please do not talk with anyone during the experiment. We ask everyone to remain silent until the end of the last decision problem.

3

Your participation in the experiment and any information about your earnings will be kept strictly confidential. Your receipt of payment and the consent form are the only places on which your name will appear. This information will be kept confidential in the manner described in the consent form. If you have any questions please ask them now. If not, we will proceed to the experiment.

4