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Chapter 2
Classic Mechanism Design
Mechanism design is the sub- eld of micro economics and game theory that considers how to implement go o d system-wide solutions to problems that involve multiple self-interested agents, each with private information ab out their preferences. In recent years mecha- nism design has found many imp ortant applications; e.g., in electronic market design, in distributed scheduling problems, and in combinatorial resource allo cation problems. This chapter provides an intro duction to the the game-theoretic approach to mechanism design, and presents imp ortant p ossibility and imp ossibility results in the literature. There is a well-understo o d sense of what can and cannot b e achieved, at least with fully rational agents and without computational limitations. The next chapter discusses the emerging eld of computational mechanism design, and also surveys the economic literature on lim- ited communication and agent b ounded-rationality in mechanism design. The challenge in computational mechanism design is to design mechanisms that are both tractable (for agents and the auctioneer) and retain useful game-theoretic prop erties. For a more general intro duction to the mechanism design literature, MasColell et al. [MCWG95 ] provides a go o d reference. Varian [Var95 ] provides a gentle intro duction to the role of mechanism design in systems of computational agents. In a mechanism design problem one can imagine that each agent holds one of the \inputs" to a well-formulated but incompletely sp eci ed optimization problem, p erhaps a constraint or an ob jective function co ecient, and that the system-wide goal is to solve the sp eci c instantiation of the optimization problem sp eci ed by the inputs. Consider for example a network routing problem in which the system-wide goal is to allo cate resources to minimize the total cost of delay over all agents, but each agent has private information ab out parameters such as message size and its unit-cost of delay. A typical approach
in mechanism design is to provide incentives (for example with suitable payments) to promote truth-revelation from agents, such that an optimal solution can b e computed to the distributed optimization problem. Groves mechanisms [Gro73 ] have a central role in classic mechanism design, and promise to remain very imp ortant in computational mechanism design. Indeed, Groves mechanisms have a fo cal role in my dissertation, providing strong guidance for the design of mechanisms in the combinatorial allo cation problem. Groves mechanisms solve problems in which the goal is to select an outcome, from a set of discrete outcomes, that maximizes the total value over all agents. The Groves mechanisms are strategy-proof, which means that truth- revelation of preferences over outcomes is a dominant strategy for each agent| optimal whatever the strategies and preferences of other agents. In addition to providing a robust solution concept, strategy-pro ofness removes game-theoretic complexity from each individ- ual agent's decision problem; an agent can compute its optimal strategy without needing to mo del the other agents in the system. In fact (see Section 2.4), Groves mechanisms are the only strategy-pro of and value-maximizing (or ecient) mechanisms amongst an imp ortant class of mechanisms. But Groves mechanisms have quite bad computational prop erties. Agents must rep ort complete information ab out their preferences to the mechanism, and the optimization problem| to maximize value |is solved centrally once all this information is rep orted. Groves mechanisms provide a completely centralized solution to a decentralized problem. In addition to dicult issues such as privacy of information, trust, etc. the approach fails computationally in combinatorial domains either when agents cannot compute their values for all p ossible outcomes, or when the mechanism cannot solve the centralized prob- lem. Computational approaches attempt to retain the useful game-theoretic prop erties but relax the requirement of complete information revelation. As one intro duces alter- native distributed implementations it is imp ortant to consider e ects on game-theoretic prop erties, for example the e ect on strategy-pro ofness. Here is an outline of the chapter. Section 2.1 presents a brief intro duction to game theory, intro ducing the most imp ortant solution concepts. Section 2.2 intro duces the the- ory of mechanism design, and de nes desirable mechanism prop erties such as eciency, strategy-pro ofness, individual-rationality, and budget-balance. Section 2.3 describ es the revelation principle, which has proved a p owerful concept in mechanism design theory, and
pure strategies. Example. In a single-item ascending-price auction, the state of the world (p; x) de nes the current ask price p 0 and whether or not the agent is holding the item in the provisional allo cation x 2 f 0 ; 1 g. A strategy de nes the bid b(p; x; v ) that an agent will place for every state, (p; x) and for every value v 0 it might have for the item. A b est-resp onse strategy is as follows:
bBR (p; x; v ) =
p , if x = 0 and p < v no bid , otherwise
One can imagine that a game de nes the set of actions available to an agent (e.g. valid bids, legal moves, etc.) and a mapping from agent strategies to an outcome (e.g. the agent with highest bid at the end of the auction wins the item and pays that price, checkmate to win the game, etc.) Again, avoiding unnecessary detail, given a game (e.g. an auction, chess, etc.) we can express an agent's utility as a function of the strategies of all the agents to capture the essential concept of strategic interdep endence.
Definition 2.2 [utility in a game] Let ui (s 1 ; : : : ; sI ; i ) denote the utility of agent i at the outcome of the game, given preferences i and strategies pro le s = (s 1 ; : : : ; sI ) selected by each agent. In other words, the utility, ui (), of agent i determines its preferences over its own strategy and the strategies of other agents, given its typ e i , which determines its base preferences over di erent outcomes in the world, e.g. over di erent allo cations and pay- ments. Example. In a single-item ascending-price auction, if agent 2 has value v 2 = 10 for the item and follows strategy bBR; 2 (p; x; v 2 ) de ned ab ove, and agent 1 has value v 1 and follows strategy bBR; 1 (p; x; v 1 ), then the utility to agent 1 is:
u 1 ( bBR; 1 (p; x; v 1 ); bBR ; 2 (p; x; 10); 10) =
v 1 (10 + ) , if v 1 > 10 0 , otherwise where > 0 is the minimal bid increment in the auction and agent i's utility given value vi and price p is ui = vi p, i.e. equal to its surplus.
The basic mo del of agent rationality in game theory is that of an expected utility maximizer. An agent will select a strategy that maximizes its exp ected utility, given its preferences i over outcomes, b eliefs ab out the strategies of other agents, and structure of the game.
2.1.2 Solution Concepts
Game theory provides a numb er of solution concepts to compute the outcome of a game with self-interested agents, given assumptions ab out agent preferences, rationality, and information available to agents ab out each other. The most well-known concept is that of a Nash equilibrium [Nas50 ], which states that in equilibrium every agent will select a utility-maximizing strategy given the strategy of every other agent. It is useful to intro duce notation s = (s 1 ; : : : ; sI ) for the joint strategies of all agents, or strategy pro le, and s i = (s 1 ; : : : ; si 1 ; si+1 ; sI ) for the strategy of every agent except agent i. Similarly, let i denote the typ e of every agent except i.
Definition 2.3 [Nash equilibrium] A strategy pro le s = (s 1 ; : : : ; sI ) is in Nash equi- librium if every agent maximizes its exp ected utility, for every i,
ui (si (i ); s i ( i ); i ) ui (s^0 i (i ); s i ( i ); i ); for all s^0 i 6 = si
In words, every agent maximizes its utility with strategy si , given its preferences and the strategy of every other agents. This de nition can b e extended in a straightforward way to include mixed strategies. Although the Nash solution concept is fundamental to game theory, it makes very strong assumptions ab out agents' information and b eliefs ab out other agents, and also loses p ower in games with multiple equilibria. To play a Nash equilibrium in a one-shot game every agent must have p erfect information (and know every other agent has the same information, etc., i.e. common knowledge) ab out the preferences of every other agent, agent rationality must also b e common knowledge, and agents must all select the same Nash equilibrium. A stronger solution concept is a dominant strategy equilibrium. In a dominant strategy equilibrium every agent has the same utility-maximizing strategy, for all strategies of other agents.
where ui is used here to denote the expected utility over distribution F ( ) of typ es. Comparing Bayesian-Nash with Nash equilibrium, the key di erence is that agent i's strategy si (i ) must b e a b est-resp onse to the distribution over strategies of other agents, given distributional information ab out the preferences of other agents. Agent i do es not necessarily play a b est-resp onse to the actual strategies of the other agents. Bayesian-Nash makes more reasonable assumptions ab out agent information than Nash, but is a weaker solution concept than dominant strategy equilibrium. Remaining problems with Bayesian-Nash include the existence of multiple equilibria, information asymmetries, and rationality assumptions, including common-knowledge of rationality. The solution concepts, of Nash, dominant-strategy, and Bayesian-Nash, hold in b oth static and dynamic games. In a static game every agent commits to its strategy simulta- neously (think of a sealed-bid auction for a simple example). In a dynamic game actions are interleaved with observation and agents can learn information ab out the preferences of other agents during the course of the game (think of an iterative auction, or stages in a negotiation). Additional re nements to these solution concepts have b een prop osed to solve dynamic games, for example to enforce sequential rationality (backwards induction) and to remove non-credible threats o the equilibrium path. One such re nement is sub- game p erfect Nash, another is p erfect Bayesian-Nash (which applies to dynamic games of incomplete information), see [FT91 ] for more details. Lo oking ahead to mechanism design, an ideal mechanism provides agents with a dom- inant strategy and also implements a solution to the multi-agent distributed optimization problem. We can state the following preference ordering across implementation concepts: dominant Bayesian-Nash Nash. In fact, a Nash solution concept in the context of a mechanism design problem is essentially useless unless agents are very well-informed ab out each others' preferences, in which case it is surprising that the mechanism infrastructure itself is not also well-informed.
2.2 Mechanism Design: Imp ortant Concepts
The mechanism design problem is to implement an optimal system-wide solution to a decentralized optimization problem with self-interested agents with private information ab out their preferences for di erent outcomes.
Recall the concept of an agent's type, i 2 i , which determines its preferences over di erent outcomes; i.e. ui (o; i ) is the utility of agent i with typ e i for outcome o 2 O. The system-wide goal in mechanism design is de ned with a social choice function, which selects the optimal outcome given agent typ es.
Definition 2.6 [So cial choice function] So cial choice function f : 1 : : : I! O cho oses an outcome f ( ) 2 O , given typ es = ( 1 ; : : : ; I ). In other words, given agent typ es = ( 1 ; : : : ; I ), we would like to cho ose outcome f ( ). The mechanism design problem is to implement \rules of a game", for example de ning p ossible strategies and the metho d used to select an outcome based on agent strategies, to implement the solution to the so cial choice function despite agent's self- interest.
Definition 2.7 [mechanism] A mechanism M = ( 1 ; : : : ; I ; g ()) de nes the set of strategies i available to each agent, and an outcome rule g : 1 : : : I! O , such that g (s) is the outcome implemented by the mechanism for strategy pro le s = (s 1 ; : : : ; sI ). In words, a mechanism de nes the strategies available (e.g., bid at least the ask price, etc.) and the metho d used to select the nal outcome based on agent strategies (e.g., the price increases until only one agent bids, then the item is sold to that agent for its bid price). Game theory is used to analyze the outcome of a mechanism. Given mechanism M with outcome function g (), we say that a mechanism implements so cial choice function f ( ) if the outcome computed with equilibrium agent strategies is a solution to the so cial choice function for all p ossible agent preferences.
Definition 2.8 [mechanism implementation] Mechanism M = ( 1 ; : : : ; I ; g ()) im- plements so cial choice function f ( ) if g (s 1 ( 1 ); : : : ; s I (I )) = f ( ), for all ( 1 ; : : : ; I ) 2 1 : : : I , where strategy pro le (s 1 ; : : : ; s I ) is an equilibrium solution to the game induced by M. The equilibrium concept is delib erately left unde ned at this stage, but may b e Nash, Bayesian-Nash, dominant- or some other concept; generally as strong a solution concept as p ossible. To understand why the mechanism design problem is dicult, consider a very naive mechanism, and supp ose that the system-wide goal is to implement so cial choice function
Definition 2.10 [Quasi-linear Preferences] A quasi-linear utility function for agent i with typ e i is of the form: ui (o; i ) = vi (x; i ) pi
where outcome o de nes a choice x 2 K from a discrete choice set and a payment pi by the agent. The typ e of an agent with quasi-linear preferences de nes its valuation function, vi (x), i.e. its value for each choice x 2 K. In an allo cation problem the alternatives K rep- resent allo cations, and the transfers represent payments to the auctioneer. Quasi-linear preferences make it straightforward to transfer utility across agents, via side-payments. Example. In an auction for a single-item, the outcome de nes the allo cation, i.e. which agent gets the item, and the payments of each agent. Assuming that agent i has value vi = $10 for the item, then its utility for an outcome in which it is allo cated the item is ui = vi p = 10 p, and the agent has p ositive utility for the outcome so long as p < $10.
Risk neutrality follows b ecause an exp ected utility maximizing agent will pay as much as the exp ected value of an item. For example in a situation in which it will receive the item with value $10 with probability , an agent would b e happy to pay as much as $10 for the item. With quasi-linear agent preferences we can separate the outcome of a so cial choice function into a choice x( ) 2 K and a payment pi ( ) made by each agent i:
f ( ) = (x( ); p 1 ( ); : : : ; pI ( ))
for preferences = ( 1 ; : : : ; I ). The outcome rule, g (s), in a mechanism with quasi-linear agent preferences, is decom- p osed into a choice rule, k (s), that selects a choice from the choice set given strategy pro le s, and a payment rule ti (s) that selects a payment for agent i based on strategy pro le s.
Definition 2.11 [quasi-linear mechanism] A quasi-linear mechanism M = ( 1 ; : : : ; I ; k (); t 1 (); : : : ; tI ()) de nes: the set of strategies i available to each agent; a choice rule k : 1 : : : I! K , such that k (s) is the choice implemented for strategy pro le s = (s 1 ; : : : ; sI ); and transfer rules ti : 1 : : : I! R , one for each agent i, to compute the payment ti (s) made by agent i.
Prop erties of so cial choice functions implemented by a mechanism can now b e stated separately, for b oth the choice selected and the payments. A so cial choice function is ecient if: Definition 2.12 [allo cative eciency] So cial choice function f ( ) = (x( ); p( )) is al locatively-ecient if for all preferences = ( 1 ; : : : ; I )
X^ I
i=
vi (x( ); i )
X
i
vi (x^0 ; i ); for all x^0 2 K (E )
It is common to state this as al locative eciency, b ecause the choice sets often de ne an allo cation of items to agents. An ecient allo cation maximizes the total value over all agents. A so cial choice function is budget-balanced if: Definition 2.13 [budget-balance] So cial choice function f ( ) = (x( ); p( )) is budget- balanced if for all preferences = ( 1 ; : : : ; I )
X^ I
i=
pi ( ) = 0 (BB)
In other words, there are no net transfers out of the system or into the system. Taken together, allo cative eciency and budget-balance imply Pareto optimality. A so cial-choice function is weak budget-balanced if: Definition 2.14 [weak budget-balance] So cial choice function f ( ) = (x( ); p( )) is weakly budget-balanced if for all preferences = ( 1 ; : : : ; I )
X^ I
i=
pi ( ) 0 (WBB)
In other words, there can b e a net payment made from agents to the mechanism, but no net payment from the mechanism to the agents.
2.2.2 Prop erties of Mechanisms
Finally, we can de ne desirable prop erties of mechanisms. In describing the prop erties of a mechanism one must state: the solution concept, e.g. Bayesian-Nash, dominant, etc.; and the domain of agent preferences, e.g. quasi-linear, monotonic, etc.
has only distributional information about the preferences of the other agents is at least its exp ected outside utility.
Definition 2.19 [individual rationality] A mechanism M is (interim) individual-rational if for all preferences i it implements a so cial choice function f ( ) with
ui (f (i ; i )) ui (i ) (IR)
where ui (f (i ; i )) is the expected utility for agent i at the outcome, given distributional information ab out the preferences i of other agents, and ui (i ) is the exp ected utility for non-participation. In other words, a mechanism is individual-rational if an agent can always achieve as much exp ected utility from participation as without participation, given prior b eliefs ab out the preferences of other agents. In a mechanism in which an agent can withdraw once it learns the outcome ex post IR is more appropriate, in which the agent's exp ected utility from participation must b e at least its b est outside utility for al l p ossible typ es of agents in the system. In a mechanism in which an agent must cho ose to participate b efore it even knows its own preferences then ex ante IR is appropriate; ex ante IR states that the agent's exp ected utility in the mechanism, averaged over all p ossible preferences, must b e at least its exp ected utility without participating, also averaged over all p ossible preferences. One last imp ortant mechanism prop erty, de ned for direct-revelation mechanisms, is incentive-compatibility. The concept of incentive compatibility and direct-revelation mech- anisms is very imp ortant in mechanism design, and discussed in the next section in the context of the revelation principle.
2.3 The Revelation Principle, Incentive-Compatibility, and
Direct-Revelation
The revelation principle states that under quite weak conditions any mechanism can b e transformed into an equivalent incentive-compatible direct-revelation mechanism, such that it implements the same so cial-choice function. This proves to b e a p owerful theoretic to ol, leading to the central p ossibility and imp ossibility results of mechanism design. A direct-revelation mechanism is a mechanism in which the only actions available to
agents are to make direct claims ab out their preferences to the mechanism. An incentive- compatible mechanism is a direct-revelation mechanism in which agents rep ort truthful information ab out their preferences in equilibrium. Incentive-compatibility captures the essence of designing a mechanism to overcome the self-interest of agents| in an incentive- compatible mechanism an agent will cho ose to rep ort its private information truthfully, out of its own self-interest. Example. The second-price sealed-bid (Vickrey) auction is an incentive-compatible (ac- tually strategy-pro of ) direct-revelation mechanism for the single-item allo cation problem.
Computationally, the revelation principle must b e viewed with great suspicion. Direct- revelation mechanisms are often to o exp ensive for agents b ecause they place very high demands on information revelation. An iterative mechanism can sometimes implement the same outcome as a direct-revelation mechanism but with less information revelation and agent computation. The revelation principle restricts what we can do, but do es not explain how to construct a mechanism to achieve a particular set of prop erties. This is discussed further in Chapter 3.
2.3.1 Incentive Compatibility and Strategy-Pro ofness
In a direct-revelation mechanism the only action available to an agent is to submit a claim ab out its preferences.
Definition 2.20 [direct-revelation mechanism] A direct-revelation mechanism M = ( 1 ; : : : ; I ; g ()) restricts the strategy set i = i for all i, and has outcome rule g : 1 : : : I! O which selects an outcome g ( ^ ) based on rep orted preferences ^ = ( ^ 1 ; : : : ; ^I ). In other words, in a direct-revelation mechanism the strategy of agent i is to rep ort typ e ^i = si (i ), based on its actual preferences i. A truth-revealing strategy is to rep ort true information ab out preferences, for all p os- sible preferences:
Definition 2.21 [truth-revelation] A strategy si 2 i is truth-revealing if si (i ) = i for all i 2 i. In an incentive-compatible (IC) mechanism the equilibrium strategy pro le s^ = (s 1 ;
2.3.2 The Revelation Principle
The revelation principle states that under quite weak conditions any mechanism can b e transformed into an equivalent incentive-compatible direct-revelation mechanism that im- plements the same so cial-choice function. The revelation principle is an imp ortant to ol for the theoretical analysis of what is p ossible, and of what is imp ossible, in mechanism design. The revelation principle was rst formulated for dominant-strategy equilibria [Gib73 ], and later extended by Green & La ont [GJJ77 ] and Myerson [Mye79 , Mye81 ]. One interpretation of the revelation principle is that incentive-compatibility comes for free. This is not to say that truth-revelation is easy to achieve, but simply that if an indirect-revelation and/or non-truthful mechanism solves a distributed optimization problem, then we would also exp ect a direct-revelation truthful implementation. The revelation principle for dominant strategy implementation states that any so cial choice function than is implementable in dominant strategy is also implementable in a strategy-pro of mechanism. In other words it is p ossible to restrict attention to truth- revealing direct-revelation mechanisms.
Theorem 2.1 (Revelation Principle). Suppose there exists a mechanism (direct or otherwise) M that implements the social-choice function f () in dominant strategies. Then f () is truthfully implementable in dominant strategies, i.e. in a strategy-proof mechanism.
Proof. If M = ( 1 ; : : : ; I ; g ()) implements f () in dominant strategies, then there exists a pro le of strategies s^ () = (s 1 (); : : : ; s I ()) such that g (s^ ( )) = f ( ) for all , and for all i and all i 2 i ,
ui (g (s i (i ); s i ); i ) ui (g ( s^i ; s i ); i )
for all ^si 2 i and all s i 2 i , by de nition of dominant strategy implementation. Substituting s i ( i ) for s i and s i ( ^i ) for ^si , we have:
ui (g (s i (i ); s i ( i )); i ) ui (g (s i ( ^i ; s i ( i )); i )
for all ^i 2 i and all i 2 i. Finally, since g (s^ ( )) = f ( ) for all , we have:
ui (f (i ; i ); i ) ui (f ( ^i ; i ); i )
for all ^i 2 i and all i 2 i. This is precisely the condition required for f () to b e truth- fully implementable in dominant strategies in a direct-revelation mechanism. The outcome rule in the strategy-pro of mechanism, g : 1 : : : I! O , is simply equal to the so cial choice function f ().
The intuition b ehind the revelation principle is as follows. Supp ose that it is p ossible to simulate the entire system| the bidding strategies of agents and the outcome rule | of an indirect mechanism, given complete and p erfect information ab out the preferences of every agent. Now, if it is p ossible to claim credibly that the \simulator" will implement an agent's optimal strategy faithfully, given information ab out the preferences (or typ e) of the agent, then it is optimal for an agent to truthfully rep ort its preferences to the new mechanism. This dominant-strategy revelation principle is quite striking. In particular, it suggests that to identify which so cial choice functions are implementable in dominant strategies, we need only identify those functions f () for which truth-revelation is a dominant strategy for agents in a direct-revelation mechanism with outcome rule g () = f (). A similar revelation principle can b e stated in Bayesian-Nash equilibrium.
Theorem 2.2 (Bayesian-Nash Revelation Principle). Suppose there exists a mecha- nism (direct or otherwise) M that implements the social-choice function f () in Bayesian- Nash equilibrium. Then f () is truthful ly implementable in a (Bayesian-Nash) incentive- compatible direct-revelation mechanism.
In other words, if the goal is to implement a particular so cial choice function in Bayesian-Nash equilibrium, it is sucient to consider only incentive-compatible direct- revelation mechanisms. The pro of closely follows that of the dominant-strategy revelation principle. One prob- lem with the revelation principle for Bayesian-Nash implementation is that the distribution over agent typ es must b e common knowledge to the direct-revelation mechanism, in addi- tion to the agents.
2.4 Vickrey-Clarke-Groves Mechanisms
In seminal pap ers, Vickrey [Vic61 ], Clarke [Cla71 ] and Groves [Gro73 ], prop osed the Vickrey-Clarke-Groves family of mechanisms, often simply called the Groves mechanisms, for problems in which agents have quasi-linear preferences. The Groves mechanisms are allo catively-ecient and strategy-pro of direct-revelation mechanisms. In sp ecial cases there is a Groves mechanism that is also individual-rational and satis es weak budget-balance, such that the mechanism do es not require an outside subsidy to op erate. This is the case, for example, in the Vickrey-Clarke-Groves mechanism for a combinatorial auction. In fact, the Groves family of mechanisms characterize the only mechanisms that are allo catively-ecient and strategy-pro of [GJJ77 ] amongst direct-revelation mechanisms.
Theorem 2.3 (Groves Uniqueness). The Groves mechanisms are the only al locatively- ecient and strategy-proof mechanisms for agents with quasi-linear preferences and general valuation functions, amongst al l direct-revelation mechanisms.
The revelation principle extends this uniqueness to general mechanisms, direct or oth- erwise. Given the premise that iterative mechanisms often have preferable computational prop erties in comparison to sealed-bid mechanisms, this uniquenss suggests a fo cus on iterative Groves mechanisms b ecause: any iterative mechanism that achieves al locative eciency in dominant-strategy imple- mentation must implement a Groves outcome. In fact, we will see in Chapter 7 that an iterative mechanism that implements the Vickrey outcome can have slightly weaker prop erties than those of a single-shot Vickrey scheme. Krishna & Perry [KP98 ] and Williams [Wil99] have recently proved the uniqueness of Groves mechanisms among ecient and Bayesian-Nash mechanisms.
2.4.1 The Groves Mechanism
Consider a set of p ossible alternatives, K , and agents with quasi-linear utility functions, such that ui (k ; pi ; i ) = vi (k ; i ) pi
where vi (k ; i ) is the agent's value for alternative k , and pi is a payment by the agent to the mechanism. Recall that the type i 2 i is a convenient way to express the valuation function of an agent. In a direct-revelation mechanism for quasi-linear preferences we write the outcome rule g ( ^ ) in terms of a choice rule, k : 1 : : : I! K , and a payment rule, ti : 1 : : : I! R , for each agent i. In a Groves mechanism agent i rep orts typ e ^i = si (i ), which may not b e its true typ e. Given rep orted typ es ^ = ( ^ 1 ; : : : ; ^I ), the choice rule in a Groves mechanism computes:
k ^ ( ^ ) = arg max k 2K
X
i
vi (k ; ^i ) (1)
Choice k ^ is the selection that maximizes the total rep orted value over all agents. The payment rule in a Groves mechanism is de ned as:
ti ( ^ ) = hi ( ^ i )
X
j 6 =i
vj (k ^ ; ^j ) (2.1)
where hi : i! R is an arbitrary function on the rep orted typ es of every agent except i. This freedom in selecting hi () leads to the description of a \family" of mechanisms. Di erent choices make di erent tradeo s across budget-balance and individual-rationality.
2.4.2 Analysis
Groves mechanisms are ecient and strategy-pro of:
Theorem 2.4 (Groves mechanisms). Groves mechanisms are al locatively-ecient and strategy-proof for agents with quasi-linear preferences.
Proof. We prove that Groves mechanisms are strategy-pro of, such that truth-revelation is a dominant strategy for each agent, from which allo cative eciency follows immediately b ecause the choice rule k ^ ( ) computes the ecient allo cation (1). The utility to agent i from strategy ^i is: ui ( ^i ) = vi (k ^ ( ^ ); i ) ti ( ^ ) = vi (k ^ ( ^ ); i ) +
X
j 6 =i
vj (k ^ ( ^ ); ^j ) hi ( ^ i )
