49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA
Robust synthesis in mechanism design Georgios Kotsalis
Jeff S. Shamma
Abstract— This paper considers two topics in mechanism design: fragility of optimal auctions and computationally constructive procedures for dynamic mechanisms. The first part of the paper considers the well studied topic in mechanism design of optimal auctions, i.e., auctions that produce maximal revenue. The design of an optimal auction in a general setting requires the principal to have complete knowledge of the probabilistic beliefs of the agents (bidders). This work shows that such an assumption leads to fragile designs in which a slight perturbation to these beliefs can alter the outcome of the auction significantly. We propose an alternative approach where the designer takes into account a nominal environment and provides incentives that are robust to perturbations on the beliefs of the agents. Using the theory of robust optimization, we find relevant uncertainty classes for which the proposed robust mechanisms have the same computational complexity as the nominal designs. The second part of the paper discusses dynamic mechanism design. Whereas standard mechanism design presumes a one shot scenario, dynamic mechanism design considers perpetual phenomena, e.g., the allocation of a limited resource to transient users. The dynamic mechanism problem has received considerable attention given the wealth of applications. Most work has concentrated on finding appropriate conditions according to which static designs have natural extensions to the dynamic setting. This paper considers optimal dynamic mechanisms using tools from robust control theory. In this framework the design of incentives for a desired social planner’s policy reduces to the search of a storage function, analogous to synthesis problems involving dissipation inequalities. For specific structures, optimizing the incentives reduces to solving a linear program.
I. I NTRODUCTION Game theory studies multi-agent decision problems, in which the payoffs of each player depend not only on his action, but also on the actions of the other players. The Nash equilibrium notion provides the main concept in predicting the outcome of a game [11]. In particular a Nash equilibrium is an action profile in which no player has an incentive to unilaterally deviate from his decision given the actions of the other players. In practical game theoretic situations the players are uncertain about elements of the game, such as the payoff parameters. This situation has been modeled by Harsanyi [7] as incomplete information games, also called Bayesian games and has become a standard in game theory literature, [11]. In this framework private information that is known to a player, such as his payoff parameters and beliefs about other players’ payoff functions, are encoded in a player’s type. Players have distributional information Georgios Kotsalis and Jeff Shamma are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. Email:
[email protected], and
[email protected]. Research was supported in part by the AFOSR grants FA9550-05-1-0321, FA9550-09-1-0420, and FA9550-05-1-0321
978-1-4244-7744-9/10/$26.00 ©2010 IEEE
over the type space, and it is typically assumed that this information is common knowledge. The Bayesian game framework underlies the concept of mechanism design. In mechanism design a principal seeks to maximize his own objective, and provides incentives to agents to participate, while acting truthfully, in an incomplete information game, [11]. The common prior/common knowledge assumptions of Bayesian games are inherited in the mechanism design literature, a feature that has come under repeated scrutiny [3], [9], [14]. In this paper we derive a framework according to which a principal can design a mechanism that is robust to uncertainty in the beliefs of the players. The discussion is focused around auctions, which offer a tangible paradigm of mechanisms. The standard optimal auction design model [10] provides a framework that allows a seller of a single indivisible object to maximize his expected profit when faced with buyers (agents), whose private valuations of the object (types) are only partially known to him. The design of an optimal auction boils down to solving a resource allocation mechanism problem by the seller. In particular buyers take actions by submitting bids, and the outcome is an allocation of the object as well as payments from the agents to the seller. Each buyer bids before learning the bids of the other buyers. Thus the mechanism induces a game of incomplete information, and the relevant solution concept is that of a BayesianNash equilibrium. A crucial assumption is that the seller, who designs the auction, has complete knowledge of the landscape in regards to the risk characteristics of the agents as well as their subjective probabilities about states of the world given their own valuation. The next section reviews the standard auction model. Subsequently we show how the assumption that the seller knows exactly the beliefs of the agents leads to fragile designs in which a slight perturbation to these beliefs can alter the outcome of the auction significantly. For the case of risk neutral buyers and a risk neutral seller, a tractable computational framework that enables the design of auctions robust is presented in section IV. The second part of the paper deals with dynamic mechanisms. Whereas standard mechanism design presumes a one shot scenario, dynamic mechanism design considers perpetual phenomena, e.g., the allocation of a limited resource to transient users. Most work has concentrated on finding appropriate conditions according to which static designs have natural extensions to the dynamic setting. A key contribution along these lines is in [2], where a dynamic analogue of the static VCG mechanism is presented. The dynamic VCG mechanism in [2] is derived under the assumption that the agent dynamics are coupled
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only through the action of the principal and is efficient, individually rational and incentive compatible in an ex post equilibrium. In this work we address the implementation question a of desired social planner’s policy and show how the design of incentives can be reduced to a linear program. The techniques developed for the static robust mechanism design as well as the dynamic implementation part are supported by examples. II. T HE STANDARD AUCTION MODEL There is one seller who has a single object to sell. He faces N agents, who are also called bidders or buyers, numbered 1, . . . , N . The set of buyers is represented by I = {1, . . . , N }. The set of possible private valuations, also known as types, is Θv = {θ1 , . . . , θm } ⊂ R+ , where m ≥ 2. Without loss of generality it is assumed that the sets of types of each Q buyer are equal, i.e. Θi = Θv , ∀i ∈ I. The set Θ = i∈N Θi represents Q the states of nature. Let I−i = I − {i} and define Θ−i = j∈I−i Θj . We sometimes write t = (ti , t−i ), where t ∈ Θ, ti ∈ Θi and t−i ∈ Θ−i . The seller has a subjective probability distribution πs over the P states of nature. Formally πs : Θ → R+ , s∈Θ πs (s) = 1, where πs (s = (s1 , . . . , sn )) = Pr(t1 = s1 , . . . , tn = sn ). In our model the seller assumes positive probability for every state of the world, i.e. πs (s) > 0, ∀s ∈ Θ. Having observed his own type ti ∈ Θi buyer i has a subjective probability distribution πi over the states P of the other players. Formally πi : Θi × Θ−i → R+ , s−i ∈Θ−i πi (si , s−i ) = 1, ∀si ∈ Θi , where πi (si , s−i ) = Pr(t−i = s−i |ti = si ). We do not impose the restriction that all agents and the seller share the same common probability beliefs. We do make the assumption that each player, conditioned on the signal he receives, assigns positive probability to every state of the other players, i.e. πi (si , s−i ) > 0, ∀s = (si , s−i ) ∈ Θ The tuple (Θ, πs , {πi }i∈I ) is called an information structure. The seller has knowledge of the information structure, thus he actually knows how the environment is perceived from point of view of each buyer. The preferences of the seller and the buyers are encoded in terms of utility functions (us , {ui }i∈I ). We assume that the utility functions are continuous and increasing real valued functions, us : R → R, ui : R → R, i ∈ I. The pair (us , {ui }i∈I ) is called the utility structure and is known to the seller. The information and utility structure define an auction problem AP = ((Θ, πs , {πi }i∈I ), (us , {ui }i∈I )). Invoking the “revelation principle” [8] allows one to confine the attention to truth-telling Bayesian-Nash equilibria of direct mechanisms. The seller asks each bidder i to reveal his type si ∈ Θi . Given the vector s = (s1 , . . . , sn ) ∈ Θ of announced types, each bidder pays an amount xi (s) to participate in a lottery in which he wins the object with probability pi (s). An auction is a pair A = ({pi : Θ → R}i∈I , {xi : Θ → R}i∈I ), subject to the feasibility constraints X pi (s) ≥ 0 ∀s ∈ Θ, ∀i ∈ I, pi (s) ≤ 1, ∀s ∈ Θ. (1)
The utility of each buyer if he does not enter the auction is normalized to 1. It is assumed that the seller has zero utility if he retains the object. III. O PTIMAL AUCTION AND BINDING CONSTRAINTS Given an auction problem the seller designs an auction that maximizes his expected utility while providing the buyers with incentives to participate, as well as to reveal their private information truthfully. The relevant solution concept in the associated game of incomplete information is that of a of a Bayesian-Nash equilibrium. Suppose that the buyers hold private information t ∈ Θ, buyer i announces si and the other agents announce their types truthfully. The expected utility of buyer i is X πi (ti , t−i )ci (ti , si , t−i ) Ui (ti , si ) = t−i ∈Θ−i
ci (ti , si , t−i ) = +
pi (si , t−i )ui (ti − xi (si , t−i ))
(1 − pi (si , t−i ))ui (−xi (si , t−i )).
Individual rationality (IR) requires that when player i announces his type truthfully his expected utility will be not less than by not participating in the auction at all Ui (ti , ti ) ≥ 1, ∀ti ∈ Θi , ∀i ∈ I.
(2)
Truthful revelation of the private information of each agent is a result of the Bayesian incentive compatibility (IC) constraints Ui (ti , ti ) ≥ Ui (ti , si ), ∀ti , si 6= ti ∈ Θi , ∀i ∈ I. The seller’s ex ante utility is X X Us = πs (t)us ( xi (t)). t∈Θ
(3)
(4)
i∈I
The auction design problem for the seller is to maximize the expected revenue (4) subject to IR (2), IC (3) and the feasibility constraints (1). In its full generality it is a nonlinear programming problem. In the rest of this section the focus will turn in characterizing active constraints in the above problem at an optimal solution point. Consider buyer i and suppose his type is ti ∈ Θi . Theorem 3.1: If the individual rationality constraint (2) is slack then at least one of the incentive compatibility constraints (3) are binding. Proof: There is one IR and m − 1 IC constraints to consider. The proof is obtained via contradiction. Assume that there is an optimal auction A = ({pi : Θ → R}i∈I , {xi : Θ → R}i∈I ), where the IR constraint corresponding to ti is slack, i.e. Ui (ti , ti ) > 1 and thatP all the IC constraints corresponding to t are slack, i.e. i t−i ∈Θ−i πi (ti , t−i )ci (ti , ti , t−i ) > P π (t , t )c (t , si , t−i ), ∀si 6= ti ∈ Θi . Consider i i −i i i t−i ∈Θ−i an alternate auction A˜ = ({˜ pi : Θ → R}i∈I , {˜ xi : Θ → R}i∈I ), where
i∈I
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p˜j = pj , ∀j ∈ I, x ˜j = xj , ∀j 6= i, x ˜i (si , t−i ) = xi (si , t−i ), x ˜i (ti , t−i ) = xi (ti , t−i )) + ǫ,
where si 6= ti ∈ Θ, ǫ > 0, t−i ∈ Θ−i . By continuity arguments there is an ǫ > 0 such that in the augmented auction A˜ the IR and the IC constraints are still slack. But then the augmented auction has higher ex ante utility than the original one, and thus the optimality of the latter is contradicted. Note that the term x˜i (ti , t−i ) appears also on the right hand side of IC constraints when the player has different private information than ti . An increase of such terms does not lead to violation of any of these constraints, since the right hand side will decrease. Theorem 3.1 indicates the fragility of optimal auctions, when there is uncertainty on the beliefs of the agents from the principal’s perspective. Suppose that buyer’s j IR constraint is binding at type tj , i.e. Uj (tj , tj ) = 1. Consider the generic case where there exist t˜−j , tˆ−j such that cj (tj , tj , t˜−j ) > cj (tj , tj , tˆ−j ). The principal designs the auction assuming that the player’s j beliefs correspond to πj , whereas his actual beliefs are πja 6= πj . In particular let πja (tj , t˜−j ) = πj (tj , t˜−j ) − ǫ, πja (tj , tˆ−j ) = πj (tj , tˆ−j ) + ǫ, πja (tj , t−j ) = πj (tj , t−j ), t−j 6= tˆ−j , t˜−j , for sufficiently small ǫ > 0. It is easy to verify that the IR constraint is then violated. Buyer j will drop out from the auction. A similar argument goes through when the IC constraint of agent j is binding at tj . An infinitesimal discrepancy between the actual distribution of buyer j and the one perceived by the principal during the design of the auction, may make it profitable for that particular buyer to misreport his type. Note that in our derivation of the fragility result we did not make any assumption on the risk preferences of the agents. The fragility of optimal auctions can also be demonstrated when there is uncertainty in the utility structure of the agents. IV. ROBUST MECHANISMS FOR RISK NEUTRAL AGENTS A. Methodology The previous section highlights the necessity for the principal to take into account uncertainty in beliefs for each player when designing an auction. In the following we consider the case of a risk neutral seller and risk neutral agents. The objective is to achieve robustness from the perspective of the principal. A robust auction problem is denoted by APr = ((Θ, πs , {Πi }i∈I ), (us , {ui }i∈I )). It distinguishes itself by its nominal counterpart through the presence of {Πi }i∈I . Those are sets of possible conditional probability distributions of the agents as opposed to singletons corresponding to the nominal case. The principal needs to ensure that the IR and IC constraints are satisfied for every agent i for every element in Πi . For specific uncertainty classes the robust auction design can be accomplished by robust optimization techniques, [1]. We first review the the nominal auction problem in the case of risk neutral actors. When the agents are risk neutral, the utility of an agent not participating in the auction is normalized to zero rather than one. The utility functions reduce to identity maps. The auction problem in the nominal case reduces to a linear program, as is verified below. Suppose that the buyers hold private information t ∈ Θ,
buyer i announces si and the other agents announce their types truthfully. The expected utility of buyer i is X πi (ti , t−i )ci (ti , si , t−i ) (5) Ui (ti , si ) = t−i ∈Θ−i
ci (ti , si , t−i )
= pi (si , t−i )ti − xi (si , t−i ).
Note that the above expression is linear in the decision variables (p, x). The seller’s ex ante utility is accordingly X X Us = πs (t)( xi (t)). (6) t∈Θ
i∈I
The principal’s objective also is linear in the decision variables. The nominal auction design problem is to maximize (6), subject to the feasibility (1), IR (2) and IC (3) constraints. Because of (5) this optimization is a linear program. The above linear program can be written in standard form. Let |Θ| = mN . Since Θ is a finite set, it can be enumerated using a one to one map h : {1, . . . , |Θ|} → Θ. The decision variables (p, x) of the auction design problem are written in a vector ζ T = (pTv , xTv ), where pTv = (p1 (h(1)), . . . , p1 (h(|Θ|)), . . . , pN (h(1)), . . . , pN (h(|Θ|)), xTv = (x1 (h(1)), . . . , x1 (h(|Θ|)), . . . , . . . , xN (h(|Θ|)). The auction design problem can then be written as max cTv ζ,
(7)
subject to ARC ζ AF ζ
≤ ≤
0, 0.
(8) (9)
The objective (7) corresponds to (6), the IR and IC constraints are lumped in (8), and the feasibility constraints are in (9). When the principal is faced with uncertainty in the agents beliefs, the constraint matrix ARC is said to belong to an uncertainty class U and (8) is replaced by ARC ζ ≤ 0, ∀ARC ∈ U. The choice of U balances the need to realistically reflect the given situation at hand as well as tractability requirements. In this work we assume that the uncertainty set is parameterized in an affine way. In particular one can write the beliefs of agent i in vector form πiv = (πi (h(1)), . . . , πi (h(|Θ|))), and P the corresponding li δ ψj πi,j , ψ∈ uncertainty set as Πi = {πiv |πiv = πi0 + j=1 0 1×|Θ| Ψ}. The vector πi ∈ R corresponds to the nominal δ distribution, πi,j ∈ R1×|Θ| is a vector of shifts, li is a vector positive integer, ψ = (ψ1 , . . . , ψli ) is the perturbation Q li [−1, 1]. This taking values in the perturbation set Ψ = j=1 affine uncertainty description can accommodate situations where the values of the beliefs of an agent are known to lie within an interval. The advantage of assuming this particular uncertainty description in the beliefs of the agents is that the robust auction design problem is still a linear program. B. Example The nominal auction design in this example is taken from [10]. There are two bidders, each of whom having a value estimate (type) ti = 10 or ti = 100 for the auctioned object. The joint probability distribution for the
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types is: Pr(10, 10) = Pr(100, 100) = 13 , Pr(10, 100) = Pr(100, 10) = 61 . Obviously the types of the agents are not independent. Solving the linear program corresponding to the nominal auction gives a revenue maximizing mechanism (p, x), where: p(10, 10) = (0.5, 0.5), x(10, 10) = (−10, −10), p(100, 100) = (0.5, 0.5), x(100, 100) = (50, 50), p(10, 100) = (0, 1), x(10, 100) = (30, 100), p(100, 10) = (1, 0), x(100, 100) = (50, 50). The expected revenue is 70. Suppose a bidder 1 is of type t1 = 10. According to the mechanism above he has to pay 30 if the other bidder’s value is t2 = 100 and expects to receive 15 if the other bidder’s value is t2 = 10. Clearly the IR constraint is binding. Suppose that bidder 1 has a slightly different perception of the environment than the one used by the principal to design the auction. In particular assume that bidder 1 assumes that for some ǫ > 0, Pr(t2 = 100|t1 = 10) = 13 + ǫ, Pr(t2 = 10|t1 = 10) = 23 − ǫ. In that case the proposed mechanism is fragile, since bidder 1 would drop from the auction. In the next step a robust auction design is performed using the methodology described in the previous section. We consider interval uncertainty in the agents beliefs, allowing the values of the beliefs of the agents to vary element by element ±φ around the nominal values given above. To solve the associated linear programming problem we used CVX, a package for specifying and solving convex programs [6]. The robust design is illustrated for various values of the parameter φ in the following figure. Note that when φ > 0.05 70
69.5
Expected revenue
69
68.5
68
67.5
67
66.5
0
0.01
0.02
Fig. 1.
0.03
0.04 0.05 0.06 Perturbation parameter
0.07
0.08
0.09
0.1
Robust auction design
the expected revenue is constant. This is because the auction mechanism in that case converges to an ex post equilibrium, making the allocation insensitive to the beliefs of the agents. V. I MPLEMENTATION - DYNAMIC MECHANISMS A. System model - Notation We consider an environment with private and dependent values in a discrete-time, infinite-horizon model. There are N agents. The state of the agents θt evolves dynamically over stages t = 0, 1, 2, . . .. The allocation (action) at stage t is denoted at ∈ A. The aggregate QN state of the system is θt = (θ1,t , . . . , θN,t ) ∈ Θ = i=1 Θi . It is assumed that A and Θi , i ∈ I are finite sets. Let W be a finite set, and
let f : Θ × A × W → Θ. The state dynamics are given by θt+1 = f (θt , at , wt ), where wt ∈ W is an iid random variable, independent of θt and at . For w ∈ W, let pw = Pr(wt = w). 1 The utility of agent i ∈ I = {1, 2, . . . , N } at stage t is determined by the current allocation at ∈ A, the current monetary transfer pi,t ∈ R and state variable θi,t ∈ Θi . The utility function ui of agent i is assumed to be quasi-linear in the monetary transfer: ui (at , pi,t , θi,t ) = vi (at , θi,t ) − pi,t . We assume that vi (at , θi,t ) ≥ 0, ∀i ∈ I, ∀θi,t ∈ Θi , ∀at ∈ A. The utility functions, state dynamics and initial distribution of the state are common knowledge at t = 0. At the beginning of each period t, each player i observes θi,t privately. The agent then makes a report of his private information to the principal, who then implements an action at and the payoffs for period t are realized. We assume that all agents, including the principal discount the future with the same discount factor δ, 0 < δ < 1. The principal is interested in implementing a specific policy, which can be represented as a memoryless feedback law. Formally a decision policy is denoted by π, where π : Θ → A, mapping reported types to actions. The specifics of the feedback law depend on the designers objective. A common criterion is to maximize social efficiency. In that case, the objective is P∞ PN maxπ E[ s=t δ s−t i=1 vi (as , θi,s )]. In this work we are not constrained to efficient decision policies, but we are addressing the implementation problem for arbitrary decision policies. The problem of efficient dynamic mechanisms in the case of uncoupled agent dynamics has been investigated in [2]. In order to implement a specific decision policy, the principal has to provide incentives to the agents to reveal their private information truthfully. This occurs through the use of payments, denoted by T = (T1 , . . . , TN ), where Ti : Θ → R. The dynamic mechanism is the tuple of the decision policy and the transfer payments, (π, T ). The agents report their private information according to strategies mapping their state to reports. Let σi denote the reporting strategy of agent i, and σ = (σ1 , . . . , σN ). We focus on direct mechanisms and restrict reporting strategies to be memoryless, i.e. σi : Θi → Θi . The ensuing results can be readily extended to the case where the agents consider reporting strategies that take a finite window of private information history into account, by appropriately redefining the state of the system. The possibility of randomized reporting strategies can be accommodated as well. However these two directions are not going to be presented here. The set of all possible reporting strategies of agent i is denoted by Σi , i.e. Σi = {σ|σ : Θi → Θi }, i ∈ I. The utility of a particular agent i ∈ I under policy π and reporting strategies σ P is given by Ji (θt , π, σ) = ∞ Ji (θi,t , θ−i,t , π, σi , σ−i ) = E[ s=t δ s−t vi (π(σ(θ)), θi,s )]. In the previous expression and subsequently in this paper, 1 There is no loss of generality by taking w to be iid random variable. t This model is equivalent to the stochastic transition kernel models for finite state systems. See also chapter 1 in [4].
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the expectation is taken with respect to the randomizing noise w. If the arguments σi or σ−i are omitted, then it is intended that the truthful reporting strategy is followed. The expected discounted sum of payments received by agent i under a dynamic mechanism (π, T ) is: Ti (θt , π, σ) = P∞ Ti (θi,t , θ−i,t , π, σi , σ−i ) = E[ s=t δ s−t Ti (π(σ(θ)), θi,s )]. The goal of the principal is to achieve implementation of some desirable policy in a game theoretic equilibrium, see [12]. Definition A dynamic mechanism (π, T ), is within period ex post incentive compatible if and only if at all stages t for all agents i ∈ I, for all possible types θt and for all reporting strategies σi ∈ Σi , Ji (θt , π) − Ti (θt , π) ≥ Ji (θi,t , θ−i,t , π, σi ) − Ti (θi,t , θ−i,t , π, σi ).
˜ → R+ , such that function Hσi : Θ −(µi (θ˜σi ,t ) − µ ˆi (θ˜σi ,t )) + Hσi (θ˜σi ,t ) ≤ X ˜ δ Hσi (F˜σi (θ˜σi ,t , wt ))pwt ∀θ˜σi ,t ∈ Θ. wt ∈W
Then the given dynamic mechanism (π, T ) is within period ex post incentive compatible. Proof: The above condition implies that −(µi (θ˜σi ,t ) − µ ˆi (θ˜σi ,t )) + Hσi (θ˜σi ,t ) ≤ ˜ ˜ δE[Hσi (θσi ,t+1 )|θ˜σi ,t )] ∀θ˜σi ,t ∈ Θ Invoking the law of iterated expectations, i.e. E[E[x|y]] = E[x] and by rearranging terms, one gets E[µi (θ˜σi ,t )− µ ˆi (θ˜σi ,t )] ≥ E[Hσi (θ˜σi ,t )]−δE[Hσi (θ˜σi ,t+1 )].
(10)
Definition A within period ex post incentive compatible dynamic mechanism (π, T ), is within period ex post individually rational if and only if at all times t for all agents i ∈ I, for all possible types θt , Ji (θt , π) − Ti (θt , π) ≥ 0.
The above relation can be repeated for subsequent times t + 1, . . . , T , while multiplying both sides of the inequality with δ s−t , s ∈ {t + 1, . . . , T } respectively. One gets δE[µi (θ˜σi ,t+1 ) − µ ˆi (θ˜σi ,t+1 )] ≥ ˜ δE[Hσ (θσ ,t+1 )] − δ 2 E[Hσ (θ˜σ i
i
i
i ,t+2
)],
.. .
The intuitive explanation of the above notion of incentive compatibility is, that if an agent knew all the information that is knowable at time t, the agent would want to report honestly as long as other agents do.
δ T −t E[µi (θ˜σi ,T ) − µ ˆi (θ˜σi ,T )] ≥ δ T −t E[Hσi (θ˜σi ,T )] − δ T +1−t E[Hσi (θ˜σi ,T +1 )]. Summing up the above relations gives
B. Main result Given a desired policy π is it possible for the principal to find transfer payments, that ensure incentive compatibility and individual rationality for each agent? This implementation question is addressed in this section. Consider the closed loop dynamics under policy π assuming that all agents report their types truthfully, θt+1 = f (θt , π(θt ), wt ) = F (θt , wt ). Suppose that agent i ∈ I uses reporting strategy σi ∈ Σi , the closed loop dynamics assuming all other agents are reporting truthfully are θˆσi ,t+1 = f (θˆσi ,t , π(σi (θˆi,σi ,t ), θˆ−i,σi ,t ), wt ) = Fˆσi (θˆσi ,t , wt ). The joint process based involving the closed loop dynamics θt and θˆσi ,t is denoted by θ˜σi ,t , i.e. θ˜σi ,t = (θt , θˆσi ,t ) ∈ ˜ Accordingly the closed loop dynamics for the joint Θ. process are written as θ˜σi ,t+1 = F˜σi (θ˜σi ,t , wt ). Define µi (θ˜σi ,t ) = vi (π(θt ), θi,t ) − Ti (θt ), and µ ˆi (θ˜σi ,t ) = vi (π(σi (θˆi,σi ,t ), θˆ−i,σi ,t ) − Ti (θˆσi ,t ). P In this notation, the IC ∞ condition (10) can be rewritten as E[ s=t δ s−t (µi (θ˜σi ,t ) − µ ˆi (θ˜σi ,t ))] ≥ 0. The following theorem shows, how the principal can design a transfer payment function Ti , which ensures that agent’s i utility, when he reports his type truthfully is at least as large as when he uses any other reporting strategy, while assuming that the rest of the agents report truthfully. It employs the concept of stochastic storage functions. Storage functions were introduced in [13] and play important role in robust control theory, for instance in connection with the KYP lemma, [5]. Theorem 5.1: Suppose for each agent i ∈ I there exists a payment function Ti and for each σi ∈ Σi , a nonnegative
T X
δ s−t E[µi (θ˜σi ,t ) − µ ˆi (θ˜σi ,t )] ≥
s=t
E[Hσi (θ˜σi ,t )] − δ T +1−t E[Hσi (θ˜σi ,T +1 )].
Since Hσi is a nonnegative valued function, one has E[Hσi (θ˜σi ,t )] ≥ 0 and therefore T X
δ s−t E[µi (θ˜σi ,t ) − µ ˆi (θ˜σi ,t )] ≥ −δ T −t E[Hσi (θ˜σi ,T +1 )].
s=t
˜ is finite E[Hσi (θ˜σi ,T +1 )] is uniformly Since the set Θ bounded and limT →∞ δ T +1−t E[Hσi (θ˜σi ,T +1 )] = 0, giving P∞ s−t E[µi (θ˜σi ,t ) − µ ˆi (θ˜σi ,t )] ≥ 0. The above condition s=t δ is equivalent to the within period ex post incentive compatibility condition, proving the theorem. Similarly to the incentive compatibility condition, one can check the within period ex post individual rationality condition by finding an appropriate storage function. Theorem 5.2: Suppose for each agent i ∈ I there exists a nonnegative function Gi : Θ → R+ , such that −(vi (π(θt ), θi,t ) − Ti (θt )) + Gi (θt ) ≤ X δ Gi (F (θt , wt ))pwt ∀θt ∈ Θ. wt ∈W
Then the given dynamic mechanism (π, T ) is within period ex post individually rational. The proof in this case is completely analogous to the proof for the incentive compatibility condition and is omitted. The above theorems can be combined so that one can establish
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the payments that are needed in order to implement a specific decision policy in a within period ex post sense. Note that the search variables are the mappings Ti , Hσi , Gi , σi ∈ Σi , i ∈ I and appear linearly in the above constraints. Thus the design of the payments can be performed by using a linear programming solver. C. Example We consider an example of processing time distribution of a server to two workstations, I = {1, 2}. This type of problem is a common abstraction for a variety of scenario’s involving allocating time to a central facility among competing agents. The server can only complete one task at a period. Delaying tasks will be discounted by the rate δ = 0.5 < 1. The principal is to sequence the jobs over time so that he maximizes the sum of discounted utilities of the agents. Let θi,t stand for the value of a task to agent i at time t. If θi,t = 0 then agent i has no task for that particular period. We take Θ1 = {0, 1} and Θ2 = {0, 2}. The allocation values are A = {1, 2}, meaning that if at = i then agent i is has access to the server. The transition dynamics are: θ1,t+1 = f1 (θ1,t , θ2,t , at , wt ) , θ2,t+1 = f2 (θ1,t , θ2,t , at , wt ). The randomizing noise takes values in W = {1, 2} and Pr(wt = 1) = 0.1, Pr(wt = 2) = 0.9. For agent 1 one has f1 (0, θ2,t , at , 1) = 1, θ2,t ∈ Θ2 , at ∈ A, f1 (0, θ2,t , at , 2) = 0, θ2,t ∈ Θ2 , at ∈ A, f1 (1, θ2,t , 1, 1) = 1, θ2,t ∈ Θ2 , f1 (1, θ2,t , 1, 2) = 0, θ2,t ∈ Θ2 , f1 (1, θ2,t , 2, wt ) = 1, θ2,t ∈ Θ2 , wt ∈ W. Similarly for agent 2 one has f2 (θ1,t , 0, at , 1) = 0, θ1,t ∈ Θ1 , at ∈ A, f2 (θ1,t , 0, at , 2) = 2, θ1,t ∈ Θ1 , at ∈ A, f2 (θ1,t , 2, 2, 1) = 2, θ1,t ∈ Θ1 , f2 (θ1,t , 2, 2, 2) = 0, θ1,t ∈ Θ1 , f2 (θ1,t , 2, 1, wt ) = 2, θ1,t ∈ Θ1 , wt ∈ W. The dynamics can be written compactly as θt = f (θt , at , wt ). The utility function for bidder i is ( θi,t , if at = i vi (at , θi,t ) = 0, if otherwise. Let Π denote the set of all decision policies, that is Π = {π | π : Θ1 × Θ2 → A}. The objective of the principal is to choose a decision that maximizes social efficiency, P∞ policy,P i.e. maxπ∈Π E[ s=t δ s−t i∈I (vi (π(θ1,t , θ2,t ), θi,t )]. The optimal cost is denoted by J ∗ : Θ → R. This is a infinite horizon, discounted, stochastic optimization problem. It can be solved using dynamic programming techniques, [4]. In particular the optimal cost is determined using value iteration. In this problem we have J ∗ (0, 0) = 1.368, J ∗(0, 2) = 2.777, J ∗(1, 0) = 2.368, J ∗(1, 2) = 3.227. The associated optimal decision policy is π ∗ (0, 0) = 1, π ∗ (0, 2) = 2π ∗ (1, 0) = 1, π ∗ (1, 2) = 2. The optimal decision policy is robust, as is, to possible misreporting by agent 1, assuming that agent 2 reports his type truthfully. Let Σi = {σ|σ : Θi → Θi }, i ∈ I. One has π ∗ (θ1 , θ2 ) = π ∗ (σ1 (θ1 ), θ2 ), ∀σ1 ∈ Σ1 . For this reason the principal does not need any payments for agent P to design s−t ∗ δ (v 1. Define Ji∗ (θt ) = E[ ∞ i (π (θ1,t , θ2,t ), θi,t )], i ∈ s=t I and recall that in P the notation of the previous section ∞ J2 (θt , π ∗ , σ2 ) = E[ s=t δ s−t (v2 (π ∗ (θ1,t , σ2 (θ2,t )), θ2,t )].
In this example, one has J2∗ (θt ) ≥ J2 (θt , π ∗ , σ2 ), ∀σ2 ∈ Σ2 . Equality is achieved, in addition to the case when the reporting strategy of agent 2 is the identity map, for the reporting strategy σ2 (θ2,t ) = 2, ∀θ2,t ∈ Θ2 . In that case π ∗ (θ1 , 2) = π ∗ (θ1 , σ2 (θ2 )) = 2 and agent 2 monopolizes the use of the facility. Though agent 2 does not gain any advantage in terms of his own derived discounted utility, he could potentially harm agent 1, who would receive then 0. By introducing transfer payments for agent 2, one can ensure that the socially efficient policy π ∗ is implemented in a within period ex post equilibrium sense and additionally J2 (θt , π ∗ ) − T2 (θt , π ∗ ) ≥ J2 (θt , π ∗ , σ2 ) − T2 (θt , π ∗ , σ2 ), with equality only when σ2 is the identity map. The stage payments are designed using the linear programming technique introduced in the previous section and are T2 (0, 0) = −0.1, T2(0, 2) = 0.1, T2(1, 0) = −0.2, T2 (1, 2) = 0.2. VI. C ONCLUSION This paper considered two topics: fragility of optimal static auctions and computationally constructive procedures for dynamic mechanism design. A framework for designing auctions that are robust to the beliefs of risk neutral agents was presented. Future work should expand this methodology to cases where the principal is faced with uncertainty both in the beliefs as well as the valuations and risk characteristics of the agents. For the dynamic problem we showed how the implementation problem of an arbitrary decision policy can be reduced to solving a linear program in order to find the appropriate incentives. The next step will be to investigate the concurrent design of decision policies and transfer payments encompassing also revenue maximization applications. R EFERENCES [1] A. Ben-Tal and A. Nemirovski. Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88:411–424, 2000. [2] D. Bergeman and J. Valimaki. The dynamic pivot mechanism. Helsinki Center of Economic Research, Discussion Paper No. 236, 2008. [3] D. Bergemann and S. Morris. Robust mechanism design. Econometrica, 73(6):1771 – 1813, 2005. [4] D. P. Bertsekas. Dynamic Programming and Optimal Control, volume 1,2. Athena Scientific, 2007. [5] G. E. Dullerud and F. Paganini. A course in robust control theory : a convex approach. Springer, 2000. [6] M. Grant and S. Boyd. Matlab software for disciplined convex programming. Web page and software, http://stanford.edu/ boyd/cvx, 2009. [7] J. Harasanyi. Games of incomplete information, played by bayesian players, parts i - iii. Management science, 14:159–182, 320–334, 486– 502, 1967-68. [8] V. Krishna. Auction theory. Academic Press, 2002. [9] J. O. Ledyard. Incentive compatibility and incomplete information. Journal of economic theory, 18(6):171 – 189, 1978. [10] R. B. Myerson. Optimal auction design. Mathematics of Operation Research, 6(1):58 – 73, 1981. [11] R. B. Myerson. Game Theory Analysis of Conflict. Harvard University Press, 1991. [12] I. Segal S. Athey. An efficient dynamic mechanism. working paper, http://kuznets.harvard.edu/ athey/., 2007. [13] J. Willems. Dissipative dynamical systems, parts i - ii. Archive for Rational Mechanics Analysis, 42:189–217, 321–351, 1972. [14] R. Wilson. Game theoretic approaches to trading processes. In T. Bewley, editor, Advances in Economic theory: Fifth world congress, pages 33–77. Cambridge University Press, 1987.
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