Utility Equivalence in Auctions - CiteSeerX

Utility Equivalence in Auctions

∗

Shlomit Hon-Snir Department of Economics and Center for Rationality and Interactive Decision Theory, The Hebrew University of Jerusalem e-mail: [email protected] May 2001

Abstract We analyze auctions from the buyers’ point of view in a (non-symmetric) independent-private-value model of valuations. It is well-known that the revenue equivalence theorem is derived for risk- neutral agents from a utility equivalence principle: For a fixed agent, for any two auction mechanisms, the difference between the utility in equilibrium functions is a constant function, if the probability of winning (in equilibrium) functions coincide. We show that this utility equivalence principle holds only for a risk-neutral agent. We generalize the definition of utility equivalence principle. The generalized principle is discussed only for pairs of auction mechanisms, which specify the same payments when the agent does not win. It is shown that the generalized principle holds if and only if the agent has constant absolute risk attitude. The (generalized) utility equivalence principle implies that an agent is indifference to any two auctions in which both, her probability to win and her expected utility at her lowest possible type ( or at any other type) coincide. The (generalized) utility equivalence principle is further generalized to a model that allows random participation. In addition, we show that this principle takes a very special form for standard auction mechanisms, and that it can be used in analyzing competition in auction design. ∗

I am grateful to Dov Monderer for his generous help and encouragement.

1

In order to prove our results we have made a comprehensive equilibrium analysis of auctions. This analysis is performed in the most general model: The agents may have any attitude toward risk, and the distribution of types may contain atoms.

1

Introduction

Most of the research in auction theory focuses on the seller perspective. The most well-known examples are the revenue equivalence theorem1 , which provides conditions under which a seller is indifferent between various auctions, and the optimal auction theorem (Myerson(1981)), which characterizes auction mechanisms that maximize the seller revenue. However, when following Myerson’s proof for the revenue equivalence theorem, it can be seen that the revenue equivalence principle follows from a utility equivalence principle for risk neutral agents, and that these two principles almost coincide. That is, the seller is indifferent between two auction mechanisms if and only if every potential buyer is indifferent between them. Matthews (1987) was the first attempt to compare auction mechanisms from the buyers point of view, when the buyers are not risk neutral. Matthews compared first- and second-price auctions, and showed relationships between monotonicity 00 (x) ) and the preferences properties of the Arrow-Pratt measure of risk aversion ( −u 0 u (x) of agents over these two auction mechanisms2 . In particular, Mattehews showed that when an agent has constant absulute risk aversion, she is indifferent between first- and second- price auctions. This theorem was generalized by Monderer and Tennenholtz (2000a) to all k-price auctions, and to agents that may have constant absolute risk attitude (CART)3 . In this paper we prove a general utility equivalence principle, that holds for agents with constant absolute risk attitude. Furthermore, 1

Vickrey (1961), Ortega-Reichert(1968), Holt (1980), Harris and Raviv (1981), Myerson(1981), Riley and Samuelson (1981). 2 Maskin and Rily(1984) discussed the revenue of the seller in first price auctions with risk averse agents. 3 An agent has constant absolute risk attitude if her Arrow-Pratt measure is a constant function. It is well-known that such an agent has a utility function, which has the form u(x) = c

1 − eλx , λ

for all x in R,

where c > 0, and λ ∈ R. By convention, u(x) = cx when λ = 0.

2

we show that this equivalence principle holds if and only if the agents have CART. We consider a seller that wishes to sell a single4 item by an auction mechanism to one of n potential buyers. We assume the (non symmetric) independent-privatevalue model of valuations5 . Each potential buyer a is characterized by her utility for money function ua , and by her valuation structure (distribution of types). The set of possible types of a is an interval [αa , β a ]. However, we assume the most general structure of distribution functions, and in particular our model treats atoms as well as atomless distributions. The auction mechanism is typically described by a set of messages, one set for each agent, and by functions (of vector of message) that define the probability of winning the object by each agent, and the expected payment functions for each agent6 . The auction mechanism together with the valuation structure define a Bayesian game – the auction game. For a fixed equilibrium profile in this game, let Qa (t), t ∈ [αa , β a ], be the probability that agent a wins the object in equilibrium given that her valuation is t, and let U a (t) be the expected utility of this agent in equilibrium. Myerson (1981) showed that, Z U a (t) =

t

z=αa

Qa (z)dz + U a (αa )

for every t ∈ [αa , β a ].

Therefore , for every auction games A and B and equilibrium profiles in this games, a such that QaA (t) = QB (t) for every t ∈ [αa , β a ], UAa (t) − UBa (t) = UAa (s) − UBa (s)

for every t, s ∈ [αa , β a ].

We refer to this result as Myerson’s utility equivalence theorem7 . We show that: 4

this work does not deal with mechanisms for selling several items –combinatorial auctions mechanisms. Lately, Krishna and Perry (1998) generalized Myerson’s utility equivalence to such auctions, keeping the assumption of risk neutrality. 5 This assumption was made in all previous works that dealt with utility (or revenue) equivalence. However, there are several issues in auction theory that have been analyzed with out the independence assumption. The first , and classical work to remove this assumption was Milgrom and Weber (1982). 6 There are two such payment functions for each agent. One function describes the payment paid by her, when she wins the item, and the other one gives her payment when she does not win. This splitting of payments is necessary when dealing with agents that are not risk neutral. 7 Note that Myerson’s theorem implies that for every two auction games A and B such that a QA (t) = QaB (t) for every t ∈ [αa , β a ] and UAa (αa ) = UBa (αa ), UAa (t) = UBa (t)

for every t ∈ [αa , β a ].

3

• Myerson’s utility equivalence theorem holds only for risk neutral buyers. We state a utility equivalence theorem, which holds for an agent if and only if she has constant absolute risk-attitude. Since the buyers are not necessarily risk neutral, we split the expected payment function U a , U a (t) = Uga (t) + Ula (t), where Uga (t), Ula (t) are the expected gain and loss utility functions. We prove that for a CART agent, a, there exists two functions: g(Qa , Ula , t) and f (t) such that f (t) > 0 for every type t, and a

U (t) =

Z t

z=αa

g(Qa , Ula , z)dz + f (t)U a (αa )

for every t ∈ [αa , β a ].

From this result we deduce our utility equivalence principle, which generalizes Myerson’s utility equivalence principle: • Let a be a CART agent. There exists a positive function h such that: For every auction games A and B and equilibrium profiles in these games, for a a a a = QB which QA and Ul,A = Ul,B , h(t)(UA (t) − UB (t)) = h(s)(UA (s) − UB (s)),

for every t, s ∈ [αa , β a ].

We further prove that : • Our utility equivalence principle holds for CART agents only. An important consequence of the utility equivalence principle is: • Let a be a CART agent. For every two auction games and associated equiliba a = Ul,B ,and UAa (αa ) = UBa (αa ), then rium profiles, if QaA = QaB , Ul,A for every t ∈ [αa , β a ].

UA (t) = UB (t),

In order to prove the utility equivalence theorem we analyze the equilibrium structure in auction games. Some of the properties which obtained have their own significance and are proved for the most general utility functions. We show that for every equilibrium in an auction game:

4

• UAa is a Liptchitz non-decreasing function. Moreover, •

n



‘

o

(UAa )0 (t) = ET −a (ua )0 t − x(ba (t), b−a ) τ (ba (t), b−a ) , where x is the winning payment function, and τ is the probability to win function.

The equilibrium strategies are derived from the above equation (see e.g, Maskin and Riley(1984) and Monderer and Tenneholtz (2000a) ). In addition, we show that • There exists a cutoff valuation t0 such that for α ≤ t ≤ t0 , U a (t) = 0, and U a is increasing in [t0 , β]. Myerson(1981) proved that for risk neutral agent, Q is non-decreasing function. We prove that this property is true for CART buyers. In section 5 we deal with standard auctions. The main characteristics of a standard auction are that the highest bid wins, losers do not pay, and the seller has a reserve price – the price of the item can not be below this reserve price. In this section we assume a symmetric model and discuss only symmetric equilibrium. The following results are proven: • The cutoff valuation of an equilibrium in a standard auction game depends only on α, β, and the reserve price. That is, the cutoff valuation does not depend on the specific equilibrium or on the utility function. • The equilibrium strategy of a CART agent in a standard auction game is non-decreasing almost everywhere with respect to the distribution of types. Applying our utility equivalence theorem to standard auctions yields the following: • A CART agent is indifferent between every two standard auction games, with the same number of potential buyers and the same reserve price. In Section 6 we extend all the previous results to a model with random participation. In the common definition of auction mechanisms, every agent considers participation in the auction. That is, each agent knows the rules of the auctions, the 5

set of buyers, and the valuations’ distribution. In this section we follow Monderer and Tennenholtz (2000c), and we model a situation in which not all agents consider participation. We assume that for each agent there exists a random variable that determines whether this agent considers participation. This random variable may be correlated with the type of the agent. In addition, any agent who considers participation may choose not to submit a bid if her type is so low that bidding is unprofitable. This model incorporates an assumption, introduced by McAfee and McMillan(1987), to the effect that a agent need not know how many other buyers will participate. All the results from the previous sections are easily generalized to this model, and therefore the statements of the results and the proofs are omitted. Section 7 deals with competition in auctions. We assume that many sellers wish to sell identical items to a set of buyers. We adopt McAfee(1993) and Monderer and Tennholtz (2000b) model - an auction selection game. An auction selection game is a two stage game. In the first stage each seller chooses an auction mechanism from a set of feasible auction mechanisms. In the second stage, the buyers are informed about the sellers choices, and each agent decides where to bid (if at all) and what to bid. This problem has been analyzed also by Peters and Severinov(1997), Peters(1997) and Burguet and Sakovics(1999). McAfee(1993),Peters and Severinov(1997) and Peters(1997), appealing to a large market hypothesis, have analyzed this problem obtaining that in equilibrium sellers set efficient mechanisms that are equivalent to zero reserve price, second-price auction. Instead, Burguet and Sakovics (1999) analyzed the imperfect competition, but they restricted the feasible auction sets to includes only second-price auctions which differ by the reserve price and they assumed risk neutral buyers. Burguet and Sakovics (1999) show that there is no symmetric pure equilibrium. In contrast, Monderer and Tennholtz (2000b) restricted the feasible auction sets to includes only k-price auctions, and assumed that the buyers are CART buyers. Monderer and Tennholtz (2000b) showed that if a subgame perfect equilibrium, with buyers symmetric continuous strategies, exists then all the sellers choose the same mechanism and each agent randomizes among the various sites with equal probability. We analyze the subgame perfect equilibrium of the auction selection game, with CART buyers, when the feasible mechanisms set include all the standard auctions with the same reserve price. We prove, 6

• If a subgame perfect equilibrium of an auction selection game with a buyers’ symmetric equilibrium exists, each agent randomizes among the various sites with equal probabilities.

2

Preliminaries

This section presents the basic notations and assumptions that will be used to describe the auction environment throughout the paper. This environment includes a single owner (a seller), who wishes to sell an item to one of n ≥ 1 agents (potential buyers) through an action mechanism.

2.1

The agents description

The set of agents is denoted by N . We assume N = {1, 2, ..., n}, n ≥ 1. Every agent a has a Von-Neumann Morgenstern utility function for money, ua (x), −∞ < x < ∞, such that • ua (0) = 0. • ua is twice differentiable. • (ua )0 (x) > 0 for every x ∈ R, where R denotes the set of the real numbers. Throughout the paper, whenever possible, we will omit the agent superscript. In this paper we will mainly deal with agents who have constant absolute risk attitude (CART ). We refer to such an agent as a CART agent. The set of all utility functions of the CART agents is denoted by CART . Recall that an agent 00 (x) has a CART if and only if there exists a constant λ such that uu0 (x) = λ for all x. Note that, for such agent, the Arrow-Pratt measure of risk attitude is constant and it is −λ. If λ = 0 the agent is risk neutral and the utility function has the form u(x) = cx for some c > 0. If λ < 0, the agent has constant absolute risk aversion and u(x) = c(1 − eλx ), c > 0. If λ > 0, the agent has constant absolute risk seeking and u(x) = c(eλx − 1), c > 0. The following is a useful characterization of CART .

7

Lemma 1 u ∈ CART if and only if there exists Γ ∈ R such that u(a + b) = u(a) + u(b) + Γu(a)u(b) ∀a, b.

(2.1)

Proof : If u ∈ CART , then (2.1) is satisfied by Γ = λ/u0 (0). Suppose there exists Γ such that (2.1) is satisfied. Differentiating both sides of (2.1) with respect to a yields: u0 (a + b) = u0 (a)[1 + Γu(b)] ∀b. Differentiating both sides again according to a yields: u00 (a + b) = u00 (a)[1 + Γu(b)] ∀b. By dividing the two equations (note that (u)0 (x) > 0 for every x ∈ R) we get that u00 (x)/u0 (x) is constant. Hence, u ∈ CART We will use the following equality derived from the proof of Lemma 1: u0 (a) = u0 (0)[1 + Γu(a)] ∀a.

(2.2)

We proceed to discuss the agents valuations. We use the (non-symmetric) independentprivate-value model. In this model, every agent a ∈ N knows her own valuation (type, willingness to pay), ta ∈ T a , where T a = [αa , β a ], 0 ≤ αa ≤ β a . This valuation is a realization of a random variable Z˜ a which takes values in T a and has Qn a distribution function F a 8 . Let F (t) = a=1 F a (ta ) be the common distribution function on T = ×na=1 T a . The triplet (N, T, F ) is called a valuation structure.

2.2

The auction mechanism

The auction mechanism comprises of sets of messages, one for each agent, as well as rules that determine the winner and the payments. An agent a ∈ N has a message set M a that contains a message ea that is called a null message. Such a message is never actually sent, but if a does not send any actual message, the seller relates to it as if a sent ea . Let M = ×a∈N M a be the set of vector of messages, and let e ∈ M be the vector of null messages. We assume that That is, F a (ta ) = P rob(˜ z a ≤ ta ). Note that our model covers both a continuous and a discrete distribution of types. 8

8

• M a \ {ea } is a subset of some Euclidean space for every a ∈ N .9 Note that M a is a metric space with the natural metric of Euclidean spaces, and with agreeing that the distance between ea and a real message m is 1. Hence, M a and M have a natural Borel structure. A subset B a of M a is bounded if B a \ {ea } is bounded. The rest of the auction mechanism is defined by three functions τ : M → [0, 1]n ,

x : M → t, we get U (s) − U (t) ≥ 0 and therefore u is non-decreasing. Assume now that Q(t) > 0. As the integrand in (3.3) is positive for every t−a ∈ T −a , U (s) − U (t) > 0. We show that U is a Liptchitz function: Given t, to bid ba (t) is at least good as ba (s). Therefore, by plugging in (3.2) ba (s) instead of ba (t) we get, analogously to the way we got (3.3): u s − x(ba (s), b−a ) − u t − x(ba (s), b−a )

11

o

τ (ba (s), b−a ) . (3.4)

− .

As u itself is a Liptchitz function on bounded intervals, and Qa is bounded, there exists a constant C > 0 such that, U (s) − U (t) ≤ C(s − t), and hence |U (s) − U (t)| ≤ C|s − t|,

for all s, t ∈ T a .

We proceed to prove (3.1). As U is a Liptchitz function U 0 (t) exists Borel almost everywhere in T a . Let U 0 (t) exists at t, αa < t < β a . Let s > t, by (3.3) ET −a {[u (s − x(ba (t), b−a )) − u (t − x(ba (t), b−a ))] τ (ba (t), b−a )} U (s) − U (t) ≥ . s−t s−t (3.5) 0 The limit of the left-hand-side of (3.5) when s → t is U (t). On the other hand by Lebesqe Converges Theorem the right-hand-side of (3.5) converges to ET −a {u0 (t − x(ba (t), b−a )) τ (ba (t), b−a )} . That is, n



‘

o

n



‘

o

n



‘

o

U 0 (t) ≥ ET −a u0 t − x(ba (t), b−a ) τ (ba (t), b−a ) . Similarly, by (3.4) we get: U 0 (t) ≤ ET −a u0 t − x(ba (t), b−a ) τ (ba (t), b−a ) . Therefore

U 0 (t) = ET −a u0 t − x(ba (t), b−a ) τ (ba (t), b−a ) . Since U is a Liptchitz function, U is a continuous function , and in particular, there exists cutoff valuation which till it the expected utility is zero and from it on the expected utility is positive. Cutoff Valuation–Definition: Let A be an auction game with an equilibrium profile b. The cutoff valuation of agent a in (A, b) is the maximal type t ∈ T a for which UAa (t) = 0. If UAa (t) > 0 for every t, the cutoff valuation is defined as α. The cutoff valuation is denoted by t0 = t0a (A, b). Note that if t0 > α, U a (t0 ) = 0. The next lemma exhibits the relation between Q, U, t0 .

12

Lemma 2 Let A be an auction game, b be a fixed equilibrium profile in A and let a be a fixed agent with the utility function u. Let t0 be the cutoff valuation of agent a in (A, b). 2.1: If QaA (s) = 0, then for every t < s, QaA (t) = 0. a 2.2: If UAa (t0 ) = 0, then for every t < t0 QaA (t) = 0, and for every t > t0 QA (t) > 0.

2.3: If UAa (t0 ) = 0, then UAa is increasing in [t0 , β a ]. That is, for every t0 ≤ t < s, UAa (t) < UAa (s). Proof : Proof of 2.1: Assume negatively, that there exists t, t < s, such that QaA (t) > 0. Hence, by Theorem 1, UAa (s) − UAa (t) > 0. On the other hand, by (3.4), UAa (s) − UAa (t) ≤ 0. A contradiction. Therefore QaA (t) = 0. Proof of 2.2: Let t be such that t < t0 and assume negatively that QaA (t) > 0. Let t < s < t0 , by Theorem 1, UAa (s) > UAa (t). On the other hand, since t < s < t0 , UAa (t) = UAa (s) = 0. A contradiction. Let t be such that t0 < t and assume negatively that QaA (t) = 0. Therefore, by (3.4), we get that UAa (t) − UAa (t0 ) ≤ 0. Since UAa (t0 ) = 0, we get UAa (t) = 0. On the other hand, t0 < t hence UAa (t) > 0. A contradiction. Proof of 2.3: Let t be such that t0 < t. By (2.2) of Lemma 2, QaA (t) > 0. Let s > t, hence by Theorem 1, UAa (t) < UAa (s). Since U is continuous, U is increasing in [to , β]. The following example shows that the condition UAa (t0 ) = 0 is necessary for (2.2) and (2.3) of Lemma 2. In this example, UAa (t0 ) > 0, and UAa (.) is constant in [to , β]. Example 1: Consider the following symmetric auction mechanism: τAa (.) = 0,

xaA (.) = 0,

yAa (.) = −4.

The expected utility is UAa (t) = ua (4), for every t in [α, β]. Therefore UAa (.) is not increasing. Thus (2.3) of Lemma 2 does not hold. a (t) = 0 for every t in [α, β], therefore (2.2) of Lemma 2 does not In addition, QA hold. 13

Lemma 3 Let A be an auction game, b be a fixed equilibrium profile in A, and let a be a fixed agent with the utility function u. Let t0 be the cutoff valuation of agent a in (A, b). If yAa (m) ≥ 0 for every m ∈ M, then UAa is increasing in [t0 , β a ]. Proof : Recall that, n 

‘



UAa (t) = ET −a u t − x(ba (t), b−a ) τ (ba (t), b−a ) + u −y(ba (t), b−a ) Since yAa (m) ≥ 0 for every m ∈ M, and t > t0 we get: n 

‘

‘h

io

1 − τ (ba (t), b−a )

o

0 < UAa (t) ≤ ET −a u t − x(ba (t), b−a ) τ (ba (t), b−a ) . Hence, P robF −a {τ (ba (t), b−a ) > 0} > 0. As the integrand in (3.3) is positive for every t−a ∈ T −a , UAa (t) < UAa (s).

a Myerson (1981) proved that for a risk neutral agent, QaA (t) ≤ QA (s), for every t < s. We generalize this result for CART agent.

Theorem 2 Let A be an auction game, b be a fixed equilibrium profile in A, and let a be a fixed CART agent with the utility function u. If yAa (m) = 0 for every m ∈ M, then QaA (t) ≤ QaA (s) for every t < s. Proof : Assume negatively, that there exists s > t such that QaA (t) > QaA (s). As ba is an equilibrium strategy, therefore given t , to bid ba (t) is at least good as bidding ba (s). That is, given t, n 

‘

n 

‘



‘

o

o

‘

n 

ET −a u t − x(ba (t), b−a ) τ (ba (t), b−a ) ≥ ET −a u t − x(ba (s), b−a ) τ (ba (s), b−a ) (3.6). Similarly, given s, o

n 

‘

ET −a u s − x(ba (s), b−a ) τ (ba (s), b−a )) ≥ ET −a u s − x(ba (t), b−a ) τ (ba (t), b−a ) (3.7). As a has CART, by Lemma 1,

o

‘



u t − x(ba (s), b−a ) = u(t − s) + [1 + Γu(t − s)]u s − x(ba (s), b−a ) . Therefore (3.6) yields n

o

n 

‘

o

u(t−s)ET −a τ (ba (t), b−a ) +[1+Γu(t−s)]ET −a u s − x(ba (t), b−a ) τ (ba (t), b−a ) ≥ 14

.

n

o

n 

‘

o

u(t−s)ET −a τ (ba (s), b−a ) +[1+Γu(t−s)]ET −a u s − x(ba (s), b−a ) τ (ba (s), b−a ) (3.8). Since s > t and Q(s) < Q(t), n

n

o

u(t − s)ET −a τ (ba (s), b−a ) > u(t − s)ET −a τ (ba (t), b−a ) Hence, from (3.8) we get:

n 

‘

o

(3.9).

o

[1 + Γu(t − s)]ET −a u s − x(ba (t), b−a ) τ (ba (t), b−a ) > n 

‘

o

[1 + Γu(t − s)]ET −a u s − x(ba (s), b−a ) τ (ba (s), b−a ) . If Γ ≤ 0, then Γu(t − s) ≥ 0. Therefore, n 

‘

o

n



‘

o

ET −a u s − x(ba (t), b−a ) τ (ba (t), b−a ) > ET −a (u s − x(ba (s), b−a ) τ (ba (s), b−a ) . (3.10) But (3.7) and (3.10) can’t hold together, therefore Q(s) ≥ Q(t). By rewriting (3.7) (using Lemma 1) we get the same result in the case of Γ ≥ 0.

4

Utility Equivalence

This paper generalizes Myerson’s utility equivalence theorem. Myerson (1981) proved that risk neutral agent is indifferent up to a constant between any two auction mechanisms which have the same probability of winning function, Q. We will prove that such a result holds only for risk neutral agents. Moreover we introduce another utility equivalence which holds if and only if the agent is a CART agent.

4.1

Myerson’s utility equivalence theorem

Myerson (1981) proved the following equivalence theorem: Theorem (Myerson 1981) Let a be a fixed risk-neutral agent. Let T a = [αa , β a ]. Then the following holds: Let A and B be two auction games in which the set of types of a is T a , and let b and d be fixed equilibrium profiles in A and B respectively. If a (t) = QaB (t) for Borel almost every t ∈ T a , QA 15

then, UAa (t) − UBa (t) = UAa (s) − UBa (s),

(4.1)

for every t, s ∈ T a . We proceed to show that Myerson’s equivalence principle holds only for risk neutral agents. Theorem 3 Let a be an agent with a utility function u, and let T a = [αa , β a ], αa < β a . If the the following condition holds, a is risk-neutral: Let A and B be two auction games, in which the set of types of a is T a , and let b and d be fixed equilibrium profiles in A and B respectively. If QaA (t) = QaB (t) for Borel almost every t ∈ T a , then (4.1) holds. Proof : We will consider auctions in which the set of agents is N = {a}. Let z be a real number. Let k be a positive integer satisfying u(−z) + (k − 1)u(1) ≥ 0.

(4.2)

let Az,k be the following direct auction mechanism; For every m ∈ M a {ea }, τ a (m) = k1 , xa (m) = z, and y a (m) = −1. Obviously, Az,k generates a truth telling auction game, in which QaAz,k (ta ) = k1 for every ta ∈ T a . Note that (4.2) is satisfied for z = 0. Hence, the auction games Az,k and A0,k satisfy QaAz,k = QaA0,k . Therefore, u(t − z) − u(t) = u(s − z) − u(s), for all t, s ∈ T a . (4.3) Recall that (4.3) holds for every real number, z, and differentiate (4.3) with respect to z to get: u0 (t − z) = u0 (s − z) for every t, s ∈ [αa , β a ], and for every −∞ < z < ∞. Hence u0 is a constant function. Therefore u(x) = cx, c > 0, for every x. A slight modification in the proof of Theorem 3 shows that this theorem holds also for a fixed set of agents. That is, given a set of agents if an agent is indifferent up to a constant between any two auctions which have the same probability to win function, then she must be a risk neutral agent. 16

4.2

The utility equivalence theorem

In order to prove the utility equivalence, Myerson (1981) proved that for risk neutral agent, for every t ∈ [αa , β a ], U a (t) =

Z t

z=αa

Qa (z)dz + U a (αa ).

We generalizes this to CART agents. Theorem 4 Let A be an auction game, b be a fixed equilibrium profile in A, and let a be a fixed CART agent with the utility function u. Then, UAa (t) = u0 (0)

Z t

0

z=αa

0

a

a eu (0)Γ(t−z) [QaA (z) − ΓUA,l (z)]dz + UAa (αa )eu (0)Γ(t−α ) .

Proof : Recall that U is a Liptchitz function, and let t ∈ T a , such that U 0 (t) exists. By (3.1), o ‘ n  U 0 (t) = ET −a u0 t − x(ba (t), b−a ) τ (ba (t), b−a ) .

Moreover, for CART agent, by (2.2), u0 (a) = u0 (0)[1 + Γu(a)]. Therefore n

n 

o

‘

o

U 0 (t) = u0 (0)[ET −a τ (ba (t), b−a ) + ΓET −a u t − x(ba (t), b−a ) τ (ba (t), b−a ) ]. Hence,

U 0 (t) = u0 (0)[Q(t) + Γ(U (t) − Ul (t))]. −u0 (0)Γt

Multiplying both sides of (4.4) by e

(4.4)

and rewriting, yields 0

0

(U (t)e−u (0)Γt )0 = u0 (0)e−u (0)Γt (Q(t) − ΓUl (t)). 0

As U (t) is a Liptchitz function, U (t)e−u (0)Γt is absolutely continuous in T a , and it is the integral of its derivative. Therefore, 0

U (t) = u0 (0)eu (0)Γt

Z t

z=α

0

0

e−u (0)Γz (Q(z) − ΓUl (z))dz + U (α)eu (0)Γ(t−α) .

(4.5)

Note that, U depends only on u,Q , Ul (.) and U (α). In addition, for risk neutral agent, U depends only on Q and U (α). The following theorem generalizes Myerson utility equivalence theorem to CART agents: 17

Theorem 5 Let a be a CART agent with the utility function u, and let T a = [αa , β a ]. Then there exists a positive function h(t), t ∈ T a such that the following holds: Let A and B be two auction games, in which the set of types of a is T a , and let b and d be fixed equilibrium profiles in A and B respectively. If QaA (t) = QaB (t) for Borel almost every t ∈ T a ,

(4.6)

a a (t) for Borel almost every t ∈ T a , UA,l (t) = UB,l

(4.7)

h(t) (UAa (t) − UBa (t)) = h(s) (UAa (s) − UBa (s)) ,

(4.8)

and then, for every t, s ∈ T a . Moreover, when u(x) = c(1 − eλx ), λ < 0, or u(x) = c(eλx − 1), λ > 0, one can choose h(t) = eλ(α−t) for every t ∈ T a . In addition, if a is risk neutral then (4.6) without (4.7) implies (4.8) with h(t) = 1 for every t ∈ T a . Proof : Let A and B be two auction games. By Theorem 4: UAa (t) = u0 (0)

Z t

a (z)]dz + UAa (αa )eu (0)Γ(t−α ) . eu (0)Γ(t−z) [QaA (z) − ΓUA,l

Z t

a a eu (0)Γ(t−z) [QB (z)]dz + UBa (αa )eu (0)Γ(t−α ) . (z) − ΓUB,l

z=αa

0

0

a

0

0

a

And, UBa (t)

0

= u (0)

z=αa

a a a (t) and UA,l Therefore, if QaA (t) = QB (t) = UB,l (t) for Borel almost every t, then, 0

UA (t) − UB (t) = eu (0)Γ(t−α) (UAa (α) − UBa (α)) for every t ∈ T a . 0

Therefore, for h(t) = eu (0)Γ(α−t) = eλ(α−t) , t ∈ T a , h(t) (UA (t) − UB (t)) = h(s) (UA (s) − UB (s))

for every t, s ∈ T a .

Finally note that if a is risk neutral, Γ = 0, and therefore h(t) = 1 and the right side of (4.5) depends on Q and U (αa ) only. One can conclude that h(t) must has the form h(t) = ceλ(α−t) , for every t ∈ T a , where c > 0. The following is an important corollary of Theorem 5: 18

Corollary 1 Let a be a fixed CART agent with the utility function u. Let A and B be two auction games with the same set of types for a, and let b and d be fixed a equilibrium profiles in A and B respectively. Assume that QaA (t) = QaB (t), UA,l (t) = a a a a a a UB,l (t) for Borel almost every t ∈ T , and UA (α ) = UB (α ). Then, UAa (ta ) = UBa (ta ) for every ta ∈ T a .

(4.9)

We proceed to prove a converse to Theorem 5. Theorem 6 Let a be a fixed agent with the utility function u and a set of types T a such that αa < β a . Assume there exists a positive function h(ta ), ta ∈ T a such that the following holds: For every two auction games A and B, in which the set of types of a is T a , and for every equilibrium profiles b and d in A and B respectively such a a a (.), that QA (.) = QaB (.) and UA,l (.) = UB,l h(ta ) (UAa (ta ) − UBa (ta )) = h(sa ) (UA (sa ) − UB (sa ))

for every t, s ∈ T a .

(4.10)

Then a is a CART agent. Proof : We consider the auctions A = Az,k and B = A0,k defined in the proof of Theorem 3. Recall that, UAz,k (t) = k1 u(t − z) + [1 − k1 ]u(1) and UA0,k (t) = 1 u(t) + [1 − k1 ]u(1). k By (4.10), (4.11) h(t) ((u(t − z) − u(t)) = h(s) (u(s − z) − u(s)) for every s, t ∈ T a = [αa , β a ], and −∞ < z < ∞. Twicely differentiating both sides of (4.11) with respect to z yields h(t)u0 (t − z) = h(s)u0 (s − z) and h(t)u00 (t − z) = h(s)u00 (s − z). 00

Therefore, uu0 is a constant function, and therefore u ∈ CART . A slight modification in the proof of Theorem 6, as in Theorem 3, shows that this theorem holds also for a fixed set of agents.

19

5

Standard auctions

In this section we provide specific conditions under which a CART agent is indifferent between two standard auctions. A standard auction is an auction mechanism in which the highest bid wins and losers don’t pay. The standard auctions constitute a significantly broad class of auctions.

5.1

A definition of standard auctions

A standard auction game is obtained from a standard auction mechanism in an agent symmetric model. A model is an agent symmetric model if all the agents have the same utility function, the same set of types and the same probability distribution over the set of types. We denote by F any F a . An anonymous mechanism auction mechanism is an auction mechanism in which all agents have the same message space, and the functions τ, x, y have the following property: For every two agents i, j ∈ N (not necessarily distinct), and for every one-to-one function π : N → N with π(i) = j and π(j) = i, τ j (mπ ) = τ i (m) for every m ∈ M , xj (mπ ) = xi (m) for every m ∈ M , and y j (mπ ) = y i (m) for every m ∈ M , where mπ is obtained from m by exchanging mi with mj . A standard auction mechanism is an anonymous auction mechanism in which:

S1: The message set of an ambiguous agent a includes all the real numbers which are greater than a minimal bid, mmin ≥ 0. That is, M a = [mmin , ∞) ∪ {ea }. S2: If at least one agent participate, then the item is given to one of the agents. P That is, i∈N τ i (m) = 1 for every m 6= e = (ei )i∈N . S3: The highest bid wins, that is: for every m ∈ M and a ∈ N, if ma 6= e and ma > mj for every j ∈ N/a such 20

6 e then τ a (m) = 1, that mj = and if τ a (m) > 0 then ma ≥ mj for every j ∈ N/a such that mj 6= e. S4: Agents who do not win, do not pay, that is y a (m) = 0 for every m ∈ M , and for every a ∈ N. S5: For every m such that τ a (m) > 0, xa (mmin , e−a ) ≤ xa (m). That is, the seller guarantee a payoff of at least xa (mmin , e−a ) for the object. We refer to xa (mmin , e−a ) as the reserve price of the seller. S6: xa (mmin , e−a ) = xa (mmin , m−a ) for every m−a ∈ M −a , for which mj ∈ {e, mmin } for every j 6= a. Assumptions S5 and S6 will be discussed at section (5.2). The most common examples of a standard auction mechanisms are k-price auctions, k ≥ 1. In a k-price auction the highest bid wins and the winner pays the k-price. That is, the winner pays the k-1 highest bid of the other agents.

• When we deal with standard auction we assume that the interval [α, β] is the minimal interval [c, d] such that P robF {c ≤ t˜ ≤ d} = 1. We focus on symmetric equilibrium; An equilibrium profile ˆb = (bi )i∈N is symmetric if bi = bj for every i, j ∈ N . For a symmetric equilibrium profile ˆb we denote by b any ba , a ∈ N , and we say that b is a symmetric equilibrium strategy.

5.2

Symmetric equilibrium in standard auction games

We prove some properties of equilibrium bid functions:

• agents do not participate when their valuations are less than the cutoff valuation. • The cutoff valuation is a property of the auction mechanism, α and β. • In equilibrium, the bid function is non-decreasing in (t0 , β]. 21

Lemma 4 Let A be a standard auction game, and let b be a symmetric equilibrium strategy with a cutoff valuation t0 > α. Then, For F a –almost every t < t0 , b(t) = e. Proof : Let S = {t ∈ [α, t0 ) : b(t) 6= e}. Assume negatively, that P robF a (S) > 0, and let t ∈ S. By (2.2) in Lemma 2, τ (b(t), b−a (t−a )) = 0 for F −a – almost every t−a ∈ T −a . Let d = Sups∈S b(s). Thus, b(t) ≤ d for every t ∈ S. For every  > 0, there exists t ∈ S with b(t) > d − . Because τ (b(t), b−a (t−a )) = 0 for F −a –almost every t−a , P robF a (z ∈ T a : b(z) ≤ d − ) = 0. Hence, b(z) = d for F a –almost every z ∈ S. Therefore, by S3, for s−a ∈ T −a such that sj ∈ S and b(sj ) = d for every j 6= a, τ (b(t), b−a (s−a )) = τ (d, d, ..., d) > 0. As P robF a (S) > 0, τ (b(t), b−a ) > 0 with a positive probability, a contradiction. Hence, P robF a (S) = 0. Next we prove that for standard auction game the cutoff valuation depends only on the reserve price of the seller, α and β. That means that all the equilibrium strategies of the same standard auction game have the same cutoff valuation. This cutoff does not change when the agents utility functions or their valuations’ structure changed (as long as the interval [α, β] is not changed). In order to proof this result we need the following two lemmas. Lemma 5 Let A be a standard auction game, and let b be a symmetric equilibrium strategy with the cutoff valuation t0 . If α < t0 ≤ β then t0 ≤ x(mmin , e−a ). Proof : Assume negatively that t0 > x(mmin , e−a ). For every  > 0 such that t0 −  > α, U (t0 − ) = 0. Therefore U (mmin |t0 − ) ≤ 0. 6 a. Note that by S3, τ (mmin , b−a (t−a )) > 0 only if bj (tj ) ∈ {e, mmin } for every j = Furthermore, by S6, x(mmin , e−a ) = x(mmin , b−a (t−a )) if bj (tj ) ∈ {e, mmin } for every j 6= a. Therefore, U (mmin |t0 − ) = ET −a 

n

‘

u(t0 −  − x(mmin , e−a ) τ (mmin , b−a ) ‘

n

o

o

= u t0 −  − x(mmin , e−a ) ET −a τ (mmin , b−a ) ≤ 0. 22

Since t0 > α, by our assumption about the minimality of the interval [α, β], P robF {α ≤ t˜ < t0 } > 0. Therefore, by lemma 3,

n

o

ET −a τ (mmin , b−a ) > 0. Hence,

‘



u t0 −  − x(mmin , e−a ) ≤ 0. Yielding

t0 −  − x(mmin , e−a ) ≤ 0. As the last inequality holds for every  > 0, t0 ≤ x(mmin , e−a ). A contradiction. Therefore t0 ≤ x(mmin , e−a ). Lemma 6 Let A be a standard auction game, and let b be a symmetric equilibrium strategy with the cutoff valuation t0 . If α ≤ t0 < β then t0 ≥ x(mmin , e−a ). Proof : Assume negatively that t0 < x(mmin , e−a ). For every  > 0 such that t0 +  < β, by S5, n 

‘

o

U (t0 + ) ≤ ET −a u t0 +  − x(mmin , e−a ) τ (b(t0 + ), b−a ) . Moreover, 0 < U (t0 + ), hence, 0 < u(t0 +  − x(mmin , e−a ))Q(t0 + ). Yielding 0 < t0 +  − x(mmin , e−a ). As the last inequality holds for every  > 0, t0 ≥ x(mmin , e−a ). A contradiction. Therefore t0 ≥ x(mmin , e−a ).

23

Theorem 7 Let A be a standard auction game, for every symmetric equilibrium strategy, 1 . If

x(mmin , e−a ) < α

2 . If

α ≤ x(mmin , e−a ) ≤ β

3 . If

then

β < x(mmin , e−a ) then

t0 = α. then

t0 = x(mmin , e−a ).

t0 = β.

Proof : Recall that the cutoff valuation ,t0 , must be in [α, β]. From Lemma 5 and Lemma 6 if α < t0 < β then t0 = x(mmin , e−a ). Therefore t0 ∈ {α, x(mmin , e−a ), β}. If x(mmin , e−a ) < α then from Lemma 5 we get t0 6= β and therefore t0 = α. If β < x(mmin , e−a ) by Lemma 6 we get t0 = β. Theorem 7 shows that t0 depends only on α, β, x(mmin , e−a ). In particular the cutoff valuation does not depend on the equilibrium strategy, the utility function, the distribution function, and the number of agents. We now show that if condition S6 is not satisfied, Theorem 7 may not be true. In this example S1-S5 satisfied. Example 2: Consider two agents, with uniform distribution on [0,1]. The auction mechanism is a modification of a symmetric second-price auction, with mmin = 0.5. If agent 2 does not participate x(m, e2 ) = 0.25 for every m ≥ 0.5. Assume that h(t) = u(t − 0.5)(0.25 − 0.5t) + u(t − 0.25)t, is increasing in t, for every t ≥ 0, where u is the agent utility function. (It is easy to find such a utility function; e.g. u(x) = x.) Note that h(0.25) < 0 and h(0.5) > 0, therefore there exists a unique t such that h(t) = 0. We denote this t by tˆ. Consider the following equilibrium strategy, b.    

e : α ≤ t < tˆ b(t) =  0.5 : tˆ < t ≤ 0.5   t : 0.5 < t ≤ β

For this equilibrium, t0 = tˆ and tˆ 6= x(mmin , e−a ). Note that in Example 2 the cutoff valuation depends on the utility function, u. 24

We now give an example of a mechanism that does not satisfy condition S5 in which for a fixed utility function, there exist two equilibrium strategies with distinct cutoff valuations. In this example S1-S4, S6 are satisfied. Example 3: Consider two agents, with uniform distribution on [0,1]. The auction mechanism is a modification of a symmetric second-price auction, with mmin = 0.5. If agent 2 does not participate x(m, e2 ) = m for every m ≥ 0.5. In addition, x(1, 1) = 0. Consider the following two equilibrium strategies, b and d.  

b(t) = 

t : 0.5 < t ≤ β e : α ≤ t ≤ 0.5

d(t) = 1 : α ≤ t ≤ β The two equilibrium strategies have distinct cutoff valuations: t0 (b) = 0.5 and t0 (d) = 0. In order to prove utility equivalence, for a fixed agent, between two standard auction games, A and B, we need to prove that QA (t) = QB (t) Borel almost everywhere. If U (t0 ) > 0 than t0 = α, otherwise, for every t < t0 , from (2.2) in Lemma 2, Q(t) = 0. Therefore if the two mechanisms have the same x(mmin , e−a ) then for every t < t0 , QA (t) = QB (t) = 0. We discuss the case, t > t0 . An equilibrium strategy, b is F a – almost everywhere non-decreasing in [t0 , β] if for every t > t0 , the following conditions hold : 1. P robFa {t < s ≤ β,

6 e, b(s) =

b(s) < b(t)} = 0.

2. P robaF {t0 ≤ s < t,

b(s) 6= e,

b(s) > b(t)} = 0.

An equilibrium strategy, b is F a – almost everywhere increasing in [t0 , β] if for every t > t0 , the following conditions hold : 1. P robaF {t < s ≤ β,

b(s) 6= e, 25

b(s) ≤ b(t)} = 0.

2. P robaF {t0 ≤ s < t,

b(s) = 6 e,

b(s) ≥ b(t)} = 0.

Theorem 8 Let A be a standard auction game with CART agents, and b be a symmetric equilibrium strategy. Then b is F a – almost everywhere non-decreasing in [t0 , β]. Proof : Assume negatively that b is not F a – almost everywhere non-decreasing in [t0 , β]. Case 1: Condition 1 does not hold. That is, there exists t > t0 , such that P robaF {S} > 0 where S = {s : t < s ≤ β, b(s) 6= e, b(s) < b(t)}. Therefore there exists s˜ ∈ S such that P robaF {s ∈ S : b(˜ s) ≤ b(s) < b(t)} > 0. a s)} = P robaF {S}.) (Otherwise for every s˜ ∈ S, P robF {s ∈ S : b(s) ≤ b(˜ Since b(˜ s) < b(t), Q(˜ s) ≤ Q(t). s) ≥ Q(t). On the other hand, Q is non-decreasing, hence Q(˜ Therefore Q(˜ s) = Q(t). A contradiction to the fact that P robaF {s ∈ S, b(˜ s) ≤ b(s) < b(t)} > 0. In the other case, condition 2 does not hold, we get a similar contradiction. Recall that if b is F a – almost everywhere increasing in [t0 , β] then for every t > t0 such that P robFa (t˜ = t) = 0, Q(t) = [F a (t)]n−1 . Now we can state two utility equivalence theorems between standard auctions. Theorem 9 Let A be a standard auction game with CART agents, and let b and d be two fixed symmetric equilibrium strategies in A. Assume that b and d are F a – almost everywhere increasing in [t0 , β]. In addition, assume that one of the following conditions hold: 1.

x(mmin , e−a ) ≥ α.

2. x(mmin , e−a ) < α

and F a (α) = 0.

Then, UAb (t) = UAd (t) for every t ∈ T a .

26

Proof : By Theorem 7, the cutoff valuation of b and d coincide, that is t0 = t0b = t0 d . Moreover, since b and d are F a – almost everywhere increasing Qb (t) = Qd (t) for Borel almost every t. We show that Ub (α) = Ud (α) = 0. If x(mmin , e−a ) ≥ α then t0 ≥ α and therefore Ub (α) = Ud (α) = 0. Otherwise, F a (α) = 0, and then Qb (α) = Qd (α) = 0 because b and d are increasing. Hence, Ub (α) = Ud (α) = 0. Therefore, Ub (α) = Ud (α) and Qb (t) = Qd (t) for Borel almost every t. By Corollary 1, UAb (t) = UAd (t) for every t ∈ T a . The proof of the following theorem is similar to the proof of Theorem 9, and therefore it is omitted. Theorem 10 Let A and B be two standard auction games with the same set of CART agents, and let b and d be fixed symmetric equilibrium strategies in A and B respectively. In addition, assume that b and d are F a – almost everywhere increasing in [t0 , β]. Assume that at least one of the following conditions hold: 1. 2.

xA (mmin , e−a ) = xB (mmin , e−a ) ≥ α.

xA (mmin , e−a ), xB (mmin , e−a ) ≤ α

and F a (α) = 0.

Then, UA (t) = UB (t) for every t ∈ T a .

5.3

Strict monotonicity

Strict monotonicity of equilibrium strategies is essential in Theorem 9 and 10. In this section we describe a class of standard auctions, in which every equilibrium strategy must be Fa – almost everywhere increasing in [t0 , β]. −a a a The payoff function x is tie-independent if xa (ma , m−a 1 ) = x (m , m2 ), for ev−a ∈ M −a , such that 0 < τ a (ma , m1−a ) < 1 and ery ma ∈ [mmin , ∞) and m−a 1 , m2 0 < τ a (ma , m−a 2 ) < 1. 27

That is, in a tie, the payoff depend only on the agent bid.

Theorem 11 Let A be a standard auction game, and b be a symmetric equilibrium strategy with the cutoff valuation t0 . If x is continuous and tie-independent, then b is Fa – almost everywhere increasing in [t0 , β]. Proof : By Theorem 8, b is Fa – almost everywhere non-decreasing in [t0 , β]. Therefore, if b is not Fa – almost everywhere increasing in [t0 , β], there exists t > t0 such that P robaF {S} > 0 where S = {s : t 6= s, b(s) = b(t)}. Let b(t) = ˆb. By tie-independence, there exists a constant c such that xa (ˆb, b−a (t−a )) = c for every t−a such that 0 < τ a (ˆb, b−a (t−a )) < 1. We show that z ≤ c, for every z ∈ S ∪ {t}. Hence, since S 6= φ and t is not in S, there exists zˆ ∈ S ∪ {t} such that zˆ < c. Let z be in S ∪ {t}. Rewrite, n 

‘o

U (z) = Ewin u z − x(ˆb, b−a )

o

n

+ Etie u (z − c) τ (ˆb, b−a ) ,

(5.1)

where win is the set of all valuations for which b(tj ) < ˆb for every j 6= a, and tie is the set of all valuations such that b(tj ) ≤ ˆb for every j 6= a and b(tj ) = ˆb for at least one agent j. For every  > 0, n 

‘o

U (ˆb +  | z) = Ewin u z − x(ˆb + , b−a ) n 

n 

‘o

+ Etie u z − x(ˆb + , b−a )

‘

o

+EW u z − x(ˆb + , b−a τ (ˆb + , b−a ) ,

+ (5.2)

6 a, ˆb < bj (tj ) ≤ ˆb + }. where W = {t−a ∈ T −a : ∃j = Since x is continuous and lim→0 P rob{W } = 0, the limit of (5.2) when  → 0 is: n 

Ewin u z − x(ˆb, b−a )

‘o

+ Etie {u (z − c)} .

As b is an equilibrium strategy, for every  > 0, U (z) ≥ U (ˆb +  | z). 28

(5.3)

That is, by (5.1) and (5.3), o

n

Etie u (z − c) [1 − τ (ˆb, b−a )] ≤ 0.

(5.4)

Recall that, the probability of the set tie is positive, moreover for every t−a ∈ tie, 0 < τ (ˆb, b−a (t−a )) < 1, therefore u(z − c) ≤ 0.

(5.5)

Hence z ≤ c, for every z ∈ S ∪ {t}. We split the proof for two cases: ˆb > mmin and ˆb = mmin . Case 1: ˆb > mmin . Let zˆ ∈ S ∪ {t} such that zˆ < c. For every  > 0, U (ˆ z ) ≥ U (ˆb −  | zˆ). Similarly to (5.4), we get o

n

z − c) τ (ˆb, b−a ) ≥ 0. Etie u (ˆ

(5.6)

u(ˆ z − c) ≥ 0.

(5.7)

Therefore, That is, zˆ ≥ c, a contradiction. Case 2: ˆb = mmin . Let zˆ ∈ S ∪ {t} such that zˆ < c. By S6, n o z − c)} + Etie u (ˆ z − c) τ (ˆb, b−a ) . U (ˆ z ) = Ewin {u (ˆ

z ) < 0, a contradiction. Therefore, U (ˆ Hence, b is Fa – almost everywhere increasing in [t0 , β]. There are other classes of standard auctions in which strict monotonicity can be proved. For examples, k-price auctions (see Monderer and Tennholtz (2000a).)

6

Random Participation

In the common definition of auction mechanisms, like the one given in the previous sections, the whole model is commonly known to all agents. That means, that every 29

agent a ∈ N consider participation in the auction. That is, agent a knows the rules of the auctions, the set of agents, and the valuations’ distribution. She may choose not to participate, but this would be a strategic decision10 . In this section we model a situation in which not all agents in N consider participation. We assume that for each agent there exists a {0, 1} random variable δ˜a such that if δ˜a = 1 this agent considers participation, and if δ˜a = 0 she does not. Let t˜a be the random variable that determines a0 s valuation. Hence an agent type is a pair (δ a , ta ). We do not assume that δ˜a and t˜a are stochastically independent. After all, it is likely that they are positively correlated. Therefore we noted the probability of δ˜a = 1 given t˜a = ta by pa (ta ), and the probability of t˜a ≤ ta given δ˜a = 1 by Ga (ta ). However we continue to assume the independent-private-value model in the sense that the random variables (δ˜a , t˜a ), a ∈ N are independent. A strategy of agent a is a Borel measurable function ba : T a → M a , which represented the bid if δ a = 1, recall that otherwise (δ a = 0) she bids ea . Every auction mechanism, A, (together with the valuation and participation structure) define a Bayesian game, which we call random participation auction game. Let b = (ba )a∈N be a fixed strategy profile in this game. Consider a fixed agent a ∈ N which considers to send the message ma , given that her type is ta . Consider also, that all the other players use their strategies in b according to their valuations and participation parameters, which are in T −a and ∆−a = ×i∈N,i6=a δ a respectively. As in the previous sections, let QaA (ma |ta ) and UAa (ma |ta ) be the probability that a is the winner and the expected utility of a respectively. More precisely, o n QaA (m|ta ) = E(∆−a ,T −a ) τ a (m, b−a ) and

n



‘



‘h

UAa (m|ta ) = E(∆−a ,T −a ) ua ta − xa (m, b−a ) τ a (m, b−a ) + ua −y a (m, b−a )

1 − τ a (m, b−a )

Where b−a is the bids vector of the other agents. For example, if δ 1 = 0 then the first item is e and otherwise it is b1 (t1 ). Recall that b is an equilibrium strategy profile if for every agent a, and for every ta ∈ T a such that pa (ta ) > 0, UAa (ba (ta )|ta ) ≥ UAa (m|ta ) 10

in most of auction theory it is further assumed that every agent must participate. In these models, the agent always have a safe bid that guarantees to her a non-negative expected utility.

30

io

.

for every m ∈ M a . In this paper we deal only with equilibrium profiles that satisfy a sort of subgame perfection condition. We required the equilibrium condition to hold for every ta ∈ T a. All the theorems from the previous sections can be slightly modified to hold in the case of random participation.

7

Competition in Auction Design.

When an object is offered by more than one seller the classical auction analysis does not work, because agents can choose where to bid and not only, how to bid. We follow McAfee(1993) and Monderer and Tennenholtz (2000b) model for competition among sellers. An auction selection game is a two stage game. There are m risk neutral sellers, m ≤ n, and every seller j offers to sell one item through an auction mechanism Aj , where all the items are identical. Assume further that every agent wishes to have only one item. The sellers can choose any auction mechanism in some class of auction mechanisms A. The sellers’ simultaneous choices of the mechanisms constitute the first stage of the auction selection game. At the second stage, the agents are told about the auction mechanisms that were chosen, every agent receives her type, decides in which auction to participate (if at all) and how much to bid in this auction. We assume sealed-bid auction mechanism that end at the same time. We deal with subgame perfect equilibrium. Note that every sub-game is defined by a vector A ≡ (Aj )m j=1 with Aj ∈ A. A strategy of agent a in the subgame A is defined by a pair (pa , ba ), where pa (t) = (paj (t))m j=1 is a probability vector; that is agent a, with type t, participates in auction j with probability paj (t). This means that an agent cannot choose not to participate. However, as each auction mechanism has the non participation option, we model an agent, who prefers not to participate as an agent who chooses the non participation option in each mechanism. The bidding function ba represents the bids of a at the various auction sites. That a is ba (t) = (baj (t))m j=1 , where bj (t) is the bid of a, with type t, in auction Aj . m and a vector P = (pa )na=1 define m auction games with Every subgame A = (Aj )j=1 random participation, (Aj (P ))m j=1 , one for each auction site. Let ((ba , pa ))a∈N be a strategy profile in the subgame A. Let Uja (m|ta ), be the expected utility of agent a at the random participation auction game Aj (P ) when she 31

−a bids m and all other agents consider participation and bid according to p−a j (t ) −a and b−a j (t ) respectively. Similarly we define Qaj (m|ta ). A strategy profile ((ba , pa ))a∈N forms an equilibrium in the subgame A, if for every agent a ∈ N and for every t ∈ T a the following hold: 1. For every auction site j, bj is an equilibrium profile in Aj (P ). 2. For all i,j such that pai (t) > 0 and pja (t) > 0,

Uia (bai (t)|t) = Uja (baj (t)|t). 3. For all i,j such that pai (t) = 0 and pja (t) > 0, Uia (mi |t) ≤ Uja (bja (t)|t) ∀mi ∈ Mi . ¿From now on, we deal with agent-symmetric model and therefore we omit the agent index. Moreover we denote the symmetric agent strategy in site j by (pj , bj ) . In addition we deal with ”continuous” valuation model. That is, • F is absolutely continuous in [α, β]. Monderer and Tennenholtz (2000b) analyzed the auction selection game under the assumption that u ∈ CART , A consists of the first- ,second-, and third-price auctions, pj and bj are continuous functions, and bj are strictly increasing. They showed that at every sub-game, there exists at most one agent-symmetric equilibrium, and that in this equilibrium each agent chooses each auction with an equal probability. That is, pj (t) = 1/m for every t ∈ (t0 , β]. We show the same result for a larger class of auctions, which includes standard auctions. Theorem 12 Let u ∈ CART , and assume that A consists of standard auctions with the same reserve price, α ≤ x(mmin , e−a) ≤ β. At every sub-game of the auction selection game, there exists at most one agent-symmetric equilibrium, (pj , bj )m j=1 , in which pj is continuous and bj is increasing in [t0 , β] for every auction site j. In this equilibrium each agent chooses each auction with equal probability. That is, pj (t) = 1/m for every t ∈ (t0 , β].

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Proof : Recall that in an agent-symmetric equilibrium of a standard auction game with CART agents, for every α ≤ t < t0 Q(t) = 0 and for every t0 < t ≤ β, Q(t) = [

Z t

z=α

0

p(z)F (z)dz + (1 −

Z β

z=α

p(z)F 0 (z)dz)]n−1 .

(7.1)

Moreover in an equilibrium of a standard auction Ul (t) = 0 for every t. In addition, since t0 = x(mmin , e−a) ≥ α, U (α) = 0. Let t ∈ (t0 , β]. We want to show that all the non-zero coordinates of p(t) = (p1 (t), ..., pm (t)) are equal. Assume pj (t) > 0 and pk (t) > 0. As pj , pk are continuous, pj (t) > 0 and pk (t) > 0 at a neighborhood of t. Therefore, an agent is indifferent among the two auctions j and k at this neighborhood. That is, Uj (s) = Uk (s) for every s at a neighborhood of t. By differentiating both sides of(4.5) (applied to the random participation case) we get (7.2) Qj (s) = Qk (s), for Borel almost every s at a neighborhood of t. By (7.1), differentiating both sides of (7.2) according to s, yields pj (s) = pk (s), for Borel almost every s at a neighborhood of t. As pj (.), pk (.) are continuous, pj (s) = pk (s), for every s at a neighborhood of t. In particular, pj (t) = pk (t). Thus, all non-zero entries of p(.) are identical and sum up to 1. Hence, pj (t) can have only finitely many values for t ∈ [t0 , β]. Since pj (.) is continuous, it must be a constant on [t0 , β]. In equilibrium it can not be that for some location pj (t) = 0 for all t > t0 and for other location, pk (t) > 0 for all t > t0 . Assume negatively, then for every t > t0 , the expected utility of agent a, if she deviates to site j is u (t − x(mmin , e−a )) . On the other hand the expected utility of agent a, at site k, is ‘

n 

o

Uk (t) = ET −a u t − xk (bk (t), bk−a ) τ (bk (t), b−a k ) .

(7.3)

By S5, n 

‘

o



‘

Uk (t) ≤ ET −a u t − xk (mmin , e−a ) τ (bk (t), bk−a ) = u t − xk (mmin , e−a ) Qk (t). Since, for every t > t0 , pj (t) = 0 and pk (t) > 0; we get: 

‘



‘

u t − x(mmin , e−a ) ≤ u t − xk (mmin , e−a ) Qk (t). 33

Recall that t0 = x(mmin , e−a ) ≥ α therefore for every t > t0 , Qk (t) = 1. This can not happen because the bid strategy in site k is F a – almost everywhere increasing. Hence, if an equilibrium exists, pj (t) = 1/m ∀t > t0 .

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[12] Monderer, D. and M. Tennenholtz (2000c), Asymptotically Optimal MultiObject Auctions for Risk-Averse Agents, Working Paper [13] Myerson, R. (1981), Optimal Auction Design, Mathematics Of Operations Research, Vol. 6, 58-73. [14] Ortega-Reichert, A. (1968), Models For Competitive Bidding Under Uncertainty, Unpublished PhD thesis. Dept. of Operations Research, Stanford University. [15] Peters, M. (1997), A Competitive Distribution Of Auctions, Review Of Economic Studies, Vol.64, 97-123. [16] Peters, M. and S. Severinov (1997), Competition Among Sellers Who Offer Auctions Instead Of Prices, Journal Of Economic Theory, Vol. 75, 141-179. [17] Riley, J. and W. F. Samuelson (1981), Optimal Auctions, American Economic Review, Vol. 71, No. 3, 381-392. [18] Vickrey, W.(1961), Counterspeculation, Auctions, And Competitive Sealed Tenders, Journal Of Finance, Vol. 16, 8-37.

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