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Chapter 3
Computational Mechanism Design
The classic mechanism design literature largely ignores computational considerations. It is common to assume that agents can reveal their complete preferences over all p ossible outcomes (the revelation principle), and that the mechanism can solve an optimization problem to select the b est outcome (e.g. the Groves mechanisms). It is useful to take a \markets-as-computation" view of computational mechanism design. Our goal is to use a market-based metho d, such as an auction, to compute a so cial-welfare maximizing outcome to a distributed problem. This markets-as-computation view has received attention in computer science for a numb er of years, and in particular within arti cial intelligence. In uential early work is the Market Oriented Programming (MOP) paradigm of Wellman [Wel93 ], which adopted economic equilibrium concepts as a technique to compute solutions to distributed optimization problems. Other classic early work includes that of Hub erman & Clearwater [HC95 ] on a market-based system for air-conditioning control, and Sandholm [San93 ] on economic-based mechanisms for decen- tralized task-allo cation amongst self-interested agents. Early motivation for the market-based approach recognized that markets can provide quite ecient metho ds to solve distributed problems, prices for example can summarize relevant information ab out agents' lo cal problems [Wel96 ]. The game-theoretic considera- tions of mechanism design were secondary to computational considerations. In recent years there has b een increasing fo cus on game-theoretic issues, at rst without much concern to computational tractability [RZ94], but later with attempts to integrate game-theoretic concerns and computational concerns [SL96, PU00b, Par01 ]. The tensions b etween classic game-theoretic solutions and tractable computational so- lutions so on b ecome evident as one considers the application of mechanisms to dicult
distributed optimization problems, such as supply-chain pro curement or bandwidth allo- cation. Here is an outline of this chapter. Section 3.1 considers the di erent computational concerns in an implemented mechanism, lo oking at computation b oth at the agent and at the mechanism infrastructure level. I consider the consequences of the revelation principle for computational mechanism design. Section 3.2 fo cuses on the Generalized Vickrey Auction, and intro duces metho ds to address b oth the cost of winner determination, the cost of complete information revelation, and to reduce communication costs. I also review work in the economics literature on mechanism design with limited communication structures.
3.1 Computational Goals vs. Game Theoretic Goals
Much of classic mechanism design is driven by the revelation principle (Section 2.3), which states informally that we only ever need to consider direct-revelation mechanisms. In a direct-revelation mechanism agents are restricted to sending a single message (the agent's preferences) to the mechanism, where that message makes a claim ab out the preferences of the agent over p ossible outcomes. The revelation principle provides a very imp ortant theoretical to ol, but is not useful in a constructive sense in dicult domains. The transformation assumed in the revelation principle from indirect mechanisms (e.g. an iterative auction) to direct-revelation mech- anisms (e.g. a sealed-bid auction) assumes unlimited computational resources, b oth for agents in submitting valuation functions, and for the auctioneer in computing the outcome of a mechanism [Led89 ]. In particular, the revelation principle assumes: | agents can compute and communicate their complete preferences | the mechanism can compute the correct outcome with complete information ab out all relevant decentralized information in the system. It can so on b ecome impractical for an agent to compute and communicate its complete preferences to the mechanism, and for the mechanism to compute a solution to the central- ized optimization problem. Direct-revelation mechanisms convert decentralized problems into centralized problems. Yet, the revelation principle do es have a vital role in the design of al l mechanisms, b oth direct and indirect, sealed-bid and iterative. The revelation principle provides focus
Approximation methods. Compute approximate outcomes based on agent strategies, and make connections b etween the accuracy of approximation and game-theoretic prop erties of the mechanism. Distributed computation. Move away from a centralized mo del of mechanism imple- mentation towards mo dels of decentralized computation to compute the outcome of a mechanism, based on information ab out agent preferences. Special cases. Identify tractable sp ecial cases of more general problems, and restrict the implementation space to those tractable sp ecial cases. Compact preference representation languages. Provide agents with expressive and compact metho ds to express their preferences, that avoid unnecessary details, make structure explicit, p erhaps intro duce approximations, and make the problem of com- puting optimal outcomes more tractable. Dynamic mechanisms. Instead of requiring single-shot direct-revelation, allow agents to provide incremental information ab out their preferences for di erent outcomes and solve easy problem instances without complete information revelation. The challenge is to make mechanisms computationally feasible without sacri cing useful game-theoretic prop erties, such as eciency and strategy-pro ofness.
3.2 Computation and the Generalized Vickrey Auction
The Generalized Vickrey Auction (GVA) is a classic mechanism with many imp ortant applications in distributed computational systems [NR01 , WWWMM01 ]. As describ ed in Section 2.4, the GVA is a strategy-pro of and ecient mechanism for the combinatorial allo cation problem, in which there are a set of items, G , and a set of agents, I , and the goal is to compute a feasible allo cation of items to maximize the total value across all agents. Agents rep ort values ^vi (S ) for each bundle S G , and the GVA computes an optimal allo cation based on rep orted values and also solves one additional problem with each agent taken out of the system to compute payments. From a computational p ersp ective the GVA presents a numb er of challenges:
Winner determination is NP-hard. Winner determination in the GVA is NP-hard, equivalent to the maximum weighted set packing problem. The auctioneer must solve the winner-determination problem once with all agents, and then once more with each agent removed from the system to compute payments. Agents must compute values for an exponential number of bund les of items. The GVA requires complete information revelation from each agent. The valuation prob- lem for a single bundle can b e hard [Mil00a], and in combinatorial domains there are an exp onential numb er of bundles to consider. Agents must communicate values for an exponential number of bund les of items. Once an agent has determined its preferences for all p ossible outcomes it must com- municate that information to the auctioneer. In addition to the network resource cost, this might b e undesirable from a privacy p ersp ective. A numb er of prop osals exist to address each of these problems, surveyed b elow. The rst problem, concerning the computational complexity of the auctioneer's winner de- termination problem, has received most attention. In comparison, the second problem, concerning the complexity on participants to determine their preferences has received considerably less attention. Exceptions include the brief discussion of bidding programs in Nisan [Nis00 ], and the recent progress that has b een made on dynamic mechanisms [WW00 , Par99 , PU00b ]. This dynamic approach includes my iBundle mechanism, and recent extensions to compute Vickrey payments. iBundle is an iterative combinatorial auction mechanism, able to compute ecient allo cations without complete information revelation from agents. The information savings follow from an equilibrium-based interactive solution concept, in which the ecient allo cation is computed in an equilibrium b etween the auctioneer and the agents. In addition to terminating without complete information revelation in realistic problem instances, agents in iBundle can compute optimal strategies without solving their complete lo cal valuation problems.
^k ( ^i ; i ) = k ^ (i ; i ) In other words, the agent would like to make the approximation algorithm select the b est p ossible outcome, given its true preferences and the announced preferences of the other agents, and this might p erhaps b e achieved by manipulating its inputs to the al- gorithm to \ x" the approximation. Truth-revelation is a dominant strategy within a Groves mechanism if and only if an agent cannot improve on the outcome computed by the mechanism's algorithm by misrepresenting its own preferences. This observation leads to useful characterizations of necessary properties for an approximation algorithm to retain strategy-pro ofness within a Groves mechanism. Tennenholtz et al. [TKDM00 ] intro duce a set of sucient (but not necessary) axioms for an approximation algorithm to retain strategy-pro ofness. The most imp ortant axiom essentially intro duces the following requirement (which the authors also refer to as \1- eciency"):
vi ( ^k (i ; ^ i ); i ) +
X j 6 =i
vj ( ^k (i ; ^ i ); ^j ) vi ( ^k ( ^i ; ^ i ); i ) +
X j 6 =i
vj ( k^ ( ^i ; ^ i ); ^j );
for all ^i 6 = i ; ^ i ; i. In words, an agent cannot improve the solution with resp ect to a particular set of inputs (i ; ^ i ) by unilaterally misrepresenting its own input i. Strategy-pro ofness follows quite naturally from this condition, given that a rational agent will only misrepresent its prefer- ences to improve the quality of the solution (for rep orted preferences from other agents and the agent's true preferences) computed by the mechanism. An interesting op en question is the degree to which these axioms restrict the eciency of an approximation algorithm, for a particular class of algorithms (e.g. constant factor worst-case approximations, etc.). Nisan & Ronen [NR00 ] take a di erent approach and de ne conditions on the range of an approximation algorithm, and require the algorithm to b e optimal in its range| a condition they refer to as maximal-in-range |for strategy-pro ofness with approximate winner-determination algorithms. The conditions are necessary and sucient. The maximal-in-range condition states that if K 0 K is the range of outcomes selected by the algorithm (i.e. k 2 K 0 implies there is some set of agent preferences for which the approximation algorithm ^k ( ) = k ), then the approximation algorithm must compute the
b est outcome in this restricted range for all inputs.
^k ( ) = max k 2K^0
X i2I
vi (k ; i )
for all 2 , and for some xed K 0 K. Intuitively, strategy-pro ofness follows b ecause the Groves mechanism with this rule implements a Groves mechanism in the reduced space of outcomes K 0. An agent cannot improve the outcome of the allo cation rule by submitting a corrupted input b ecause there is no reachable outcome of b etter quality. Nisan & Ronen partially characterize the nec- essary ineciency due to the dual requirements of approximation and strategy-pro ofness, and claim that all truthful mechanisms with approximate algorithms have \unreasonable" b ehavior, for an appropriate de nition of unreasonableness. Lehmann et al. [LOS99] consider strategy-pro of and approximate implementations for a sp ecial case of the combinatorial allo cation problem, with single-minded bidders that care only ab out one bundle of items. Perhaps surprisingly, winner-determination remains NP-hard even in this very restricted problem by reduction from weighted set packing. Lehmann et al. allows the payment rules to change from those in a Groves scheme, and prop ose a set of sucient axioms for strategy-pro ofness in their problem. The axioms apply to prop erties of the allo cation rule and the payment rule. The most imp ortant condition for strategy-pro ofness is the \critical" condition, which states that an agent's payment must b e indep endent of its bid and \minimal", closely following the intuition b ehind the incentive-compatibility of the Vickrey-Clarke-Groves scheme. Lehmann et al. prop ose a greedy allo cation rule and a payment scheme that satis es their axioms, which together comprise a strategy-pro of mechanism. Extending to \double-minded" agents, the authors prove that there are no strategy-pro of payment rules compatible with their greedy allo cation metho d. Nisan & Ronen [NR01 ] present an interesting algorithmic study of mechanism design for a task allo cation problem, with a non eciency-maximizing ob jective (and therefore outside of the application of Groves mechanisms). The ob jective in the task allo cation problem is to allo cate tasks to minimize the makespan, i.e. the time to complete the nal task. Individually, each agent wants to minimize the time that it sp ends p erforming tasks. Nisan & Ronen present sucient conditions for the strategy-pro ofness of a mechanism, and consider the class of approximation algorithms that satisfy those conditions. The
The app eal functions are very complex and require a high degree of insight on the part of agents. Nisan & Ronen note that the agents themselves could b e required to compute the results of their app eal function. The mechanism therefore can b e viewed as a metho d to use decentralized computation to improve the p erformance of an approximate winner- determination algorithm. It is also suggested that agents b e given the chance to learn the characteristics of the approximation algorithm, to enable them to generate go o d app eal functions. Another idea is to integrate successful app eals progressively into the heuristic, to improve its base p erformance. Rothkopf et al. [RPH98 ] had earlier prop osed decentralized computation approaches, with \challenges" issued to agents to improve the quality of the auctioneer's solution. Brewer [Bre99 ] also prop oses a market mechanism to decentralize computation to agents.
Bounded-Rational Implementation
Returning to the concept of feasible truthfulness, there is one one sense in which b ounded- rationality can help in mechanism design. Nisan & Ronen [NR00 ] intro duce the concept of a feasible best-response and a feasible dominant action. A feasible b est-resp onse is an agent's utility-maximizing action across a restricted set of all p ossible actions, known as the agent's know ledge set. The knowledge set is a mapping from the actions of other agents to a subset of an agent's own p ossible actions. An action is then feasible dominant if it is the b est-resp onse in an agent's knowledge set for all p ossible actions of other agents. This is a very similar concept to the maximal-in-range idea intro duced as an axiom for strategy-pro ofness with approximate winner-determination algorithms. Given this concept of feasible dominance, one might design mechanisms in which the strategies that p erform b etter than truth-revelation are in small measure compared to all p ossible strategies, to make an agent require a lot of \knowledge" to have a non truth- revealing dominant strategy, or p erform a lot of computation. One can also interpret the myopic best-response strategy, adopted in my own work [Par99 , PU00a ], from the p ersp ective of a b ounded-rational agent. Certainly the assump- tion of myopia considerably simpli es an agent's problem, as it do es not need to reason ab out the e ect of its bids in the current round on future prices or on the strategies of other agents.
An interesting idea for future work is to design mechanisms that cannot b e manipulated unless an agent can solve an NP-hard computational problem; i.e. use the b ounded- ratonality of an agent to make it provably to o dicult to manipulate a mechanism.
Sp ecial-Cases and Structure
Finally, let us consider the role of tractable sp ecial-cases of winner-determination. Rothkopf et al. [RPH98 ], Nisan [Nis00] and de Vries & Vohra [dVV00 ] characterize tractable sp ecial- cases, identifying restrictions on the typ es of bundles that can receive bids and/or the typ es of valuation functions agents can express over bundles (see Section 4.5). The approach is to restrict an agent's bidding language to induce only tractable winner-determination prob- lems. Ideally, a restricted bidding language can supp ort tractable winner-determination with- out preventing agents rep orting their true valuation functions. In this case the GVA mecha- nism can b e applied without any loss in either strategy-pro ofness or eciency. However, as so on as one imp oses a restriction on agents' bids there is a risk that eciency and strategy- pro ofness will b e compromised. If an agent cannot represent its true valuation function with the restricted bidding language, then its rational strategy is to rep ort an approximate value that leads to the b est outcome for its true preferences, and force the mechanism to select the b est solution from the set reachable from the restricted range of inputs. This ability to improve the outcome through non-truthful bidding leads to a loss in strategy- pro ofness, for example b ecause the agent will now need to predict the strategies of other agents. The tradeo b etween approximate bidding languages, incentive-compatibility, and eciency app ears to have received little attention. Graphical tree representations, such as the Exp ected Utility Networks [MS99 ], allow an agent to capture indep endence structure within its preferences in much the same way as Bayes-Nets provide compact representations of conditional probabilities in suitable prob- lems. In addition to providing quite compact and natural representations for participants, these structured approaches may allow tractable winner-determination and payment rules, that exploit the structure to solve problems without explicitly computing values for indi- vidual bundles.
helpful when the iterative pro cedure terminates without complete information revelation by agents, and when an agent can provide incremental information without computing its complete valuation function. Let us consider each in turn.
Bidding Programs and High Level Bidding Languages
In cho osing a bidding language for a mechanism there is a tradeo b etween the ease with which an agent can represent its lo cal preferences, and the ease with which the mechanism can compute the outcome. Nisan [Nis00 ] describ es the expressiveness of a language, which is a measure of the size of a message for a particular family of valuation functions, and the simplicity of a language, which is a measure of the complexity involved in interpreting a language and computing values for di erent outcomes. A natural starting p oint in combinatorial auctions is the XOR bidding language, (S 1 ; p 1 ) xor (S 2 ; p 2 ), which essentially allows an agent to enumerate its value for all p ossi- ble sets of items. This bidding language is simple to interpret, in fact given a bid b in the XOR language, the auctioneer can compute the value b(S ) for any bundle in p olynomial time [Nis00]. However, this bidding language is not very expressive. An obvious example is provided with a linear valuation function, v (S ) = Px 2 S v (x). XOR bids for this valuation function are exp onential in size (explicitly enumerating the value for all p ossible bundles) [Par99 ]. In comparison, an OR bidding language (S 1 ; p 1 ) or (S 2 ; p 2 ), which states that the agent wants S 1 or S 2 or b oth, has a linear-space representation of this valuation function. Nisan observes that other combinations, such as XOR-of-OR languages and OR-of- XOR languages, allow compact representations of certain preference structures and make tradeo s across expressiveness and compactness. Nisan prop oses an OR* bidding language, which is expressive enough to b e able to represent arbitrary preferences over discrete items, and as compact a representation as b oth OR-of-XOR and XOR-of-OR representations. However, Nisan provides an example with no p olynomial-size representation even with the OR* language. The expressiveness of a bidding language, or the compactness of representations that it p ermits, b ecomes even more imp ortant when one considers the agent's underlying valuation problem. Supp ose that an agent must solve an NP-hard constrained optimization problem [P ] to compute its value for a set of items, with ob jective function g and constraints C. In the
XOR representation the agent must solve this problem [P ] once for every p ossible input S G , i.e. requiring an exp onential numb er of solutions to an NP-hard problem. Now consider an alternative bidding language, that allows the agent to send the sp eci cation of its optimization problem, i.e. [P ] = (g ; C ) directly to the auctioneer. Strategy-pro ofness is not a ected (assuming the agent can trust the mechanism to interpret this bidding language faithfully), but the agent saves a lot of value computation. In general, we might consider a language in which the agent can send a \bidding program" to the auctioneer, that the auctioneer will then execute as necessary to compute an agent's value for di erent subsets of items [Nis00 ]. This is really just the extreme limit of the revelation principle: rather than requiring an agent to solve its lo cal problem and compute its value for all p ossible outcomes, simply allow the agent to send the lo cal problem sp eci cation directly to the auctioneer. From the p ersp ective of the bidding agent this approach simpli es its valuation problem whenever the sp eci cation of its lo cal problem is simpler than actually computing its value for all p ossible outcomes. A bidding program allows an agent to feed that sp eci cation directly to the auctioneer. From the p ersp ective of the auctioneer, this is an even more centralized solution than providing a complete valuation function, and has worse-still privacy implications. The bidding program approach shifts the valuation computational burden from agents to the auctioneer. Notice for example that if the bidding program provides only \black b ox" functionality, e.g. b : 2 G^! R , the mechanism must compute b(S ) for all S G (unless other consistency rules such as free disp osal apply to an agents' values) to compute the ecient solution. However, if the bidding program, or language, provides a richer functionality| for example allowing ecient pruning \the value b(S 0 ) on all bundles S 0 S is less than b(S )"; or computing approximate values \the value of b(S ) is b etween [a; b]"; or b est-resp onse \the bundle that maximizes b(S ) p(S ) at those prices is S 1 " |then the total valuation work p erformed by the auctioneer can b e less than that required by agents with the XOR bidding language. Savings of this kind can b e realized within an algorithm that makes explicit use of these typ es of query structures. Let me outline some serious limitations of the bidding program mo del in some domains:
The sp eci cation problem can b e as dicult as the valuation problem. In particular, the assumption ab ove is that a single sp eci cation allows an agent to compute its
| ordinal information, i.e. \which bundle has highest value out of S 1 , S 2 and S 3 ?" | approximate information, i.e. \is your value for bundle S 1 greater than 100?" | best-response information, i.e. \which bundle do you want at prices p(S )?" | equivalence-set information, i.e. \is there an item that is substitutable for A?" In addition to solving realistic problem instances without complete information reve- lation, it is also imp ortant that dynamic metho ds allow an agent to resp ond to requests for information with an approximate solution to its own valuation function. Notice that in each of the preceding examples an agent can resp ond without rst computing its exact value for all bundles.
Examples: Complete Information is Not Necessary
Examples 1{3 are simple problems instances in which the optimal allo cation and the Vick- rey payments can b e computed without complete information from agents. Although there is no consideration of agent incentives at this stage, a well structured iterative auction can compute optimal outcomes without complete information from agents and provide incen- tives for agents to reveal truthful information.
Example 1. Single-item auction with 3 agents, and values v 1 = 16 ; v 2 = 10 ; v 3 = 4. The Vickrey outcome is to sell the item to agent 1 for agent 2's value, i.e. for 10. Instead of information fv 1 ; v 2 ; v 3 g it is sucient to know fv 1 10 ; v 2 = 10 ; v 3 10 g to compute this outcome.
Example 2. Consider a combinatorial auction problem in which we ask every agent for the bundle that maximizes their value. If the resp onse from each agent is non-overlapping, as illustrated in Figure 3.1 then we cam immediately compute the outcome of the GVA. The ecient allo cation is to give each agent its favorite bundle; every agent gets its value- maximizing bundle so there can b e no b etter solution. The Vickrey payments in this example are zero, intuitively b ecause there is no comp etition b etween agents. We do not need any information ab out the value of an agent for any other bundles, and we do not need even need an agent's value for its favorite bundle.
Example 3. Consider the simple combinatorial allo cation problem instance in Table 3.1, with items A, B and agents 1, 2, 3. The values of agent 1 for item B and bundle AB are stated as a b and b 15, but otherwise left unde ned. Consider the following cases:
1
(^32)
4
Figure 3.1: A simple combinatorial allo cation problem. Each disc represents an item, and the selected bundles represent the bundles with maximum value for agents 1, 2, 3 and 4. In this example this is sucient information from agents to compute the ecient solution (and the Vickrey payments).
[a < 5] In this case the GVA assigns bundle AB to agent 3, with V ^ = 15, (V 3 )^ = max[10 + a; b], so that the payment for agent 3 is pvick (3) = 15 (15 max [10 + a; b]) = max[10 + a; b]. It is sucient to know fa 5 ; b 15 ; max [10 + a; b]g to compute the outcome. [a 5] In this case the GVA assigns item B to agent 1 and item A to agent 2, with V ^ = 10 + a, (V 1 )^ = 15, and (V 2 )^ = 15. The payment for agent 1 is pvick (1) = a (10 + a 15) = 5 and the payment for agent 2 is pvick (2) = 10 (10 + a 15) = 15 a. It is sucient to know fa; b 15 g to compute the outcome.
Notice that it is not necessary to compute the value of the optimal allo cation S^ to compute Vickrey payments; we only need to compute the allo cation to each agent. Consider Example 1. We can compute the optimal allo cation (give the item to agent 1) with information v 1 fv 2 ; v 3 g, and without knowing the exact value of v 1. Also, it is not even necessary to compute V ^ and (V i )^ to compute Vickrey payments b ecause common terms cancel. In Example 1, it is enough to know the value of v 2 to compute agent 1's Vickrey payment b ecause the value of v 1 cancels: pvick (1) = v 1 vick (1) = v 1 (v 1 v 2 ) = v 2.
Useful Prop erties of Iterative Auctions
Iterative price directed auctions, such as ascending-price auctions, present an imp ortant class of dynamic mechanisms. In each round of the auction the auctioneer announces prices
auction eventually terminates in comp etitive equilibrium. iBundle solves the problem in Figure 3.1 in one round with myopic b est-resp onse agent strategies, b ecause every agent will bid for its value-maximizing bundle in resp onse to zero prices and every agent will receive a bundle in the provisional allo cation. In fact, iBundle is provably ecient with myopic b est-resp onse agent strategies.
3.2.3 Communication Costs: Distributed Metho ds
Shoham & Tennenholtz [ST01] explore the communication complexity of computing simple functions within an auction-based algorithm (i.e., with self-interested agents with private information). Essentially, the authors prop ose a metho d to compute solutions to simple functions with minimal communication complexity. Communication from the auctioneer to the agents is free in their mo del, while communication from agents to the auctioneer is costly. Given this, Shoham & Tennenholtz essentially provide incentive schemes so that each agent i announces its value vi by sending a single bit to the mechanism whenever the price in an auction is equal to this value. Max and min functions can b e computed with a single bit from agents, and any function over n agents can b e computed in n bits, which is the lower information-theoretic b ound. Feigenbaum et al. [FPS00 ] investigate cost-sharing algorithms for multicast transmis- sion, in which a p opulation of consumers sit on the no des of a multicast tree. Each user has a value to receive a shared information stream, such as a lm, and each arc in the multicast tree has an asso ciated cost. The mechanism design problem is to implement the multicast solution that maximizes total user value minus total network cost, and shares the cost across end-users. Noting that budget-balance, eciency, and strategy-pro ofness are imp ossibility in combination the authors compare the computational prop erties of a Vickrey-Clarke-Groves marginal cost (MC) mechanism (ecient and strategy-pro of ) and a Shapley value (SH) mechanism (budget-balanced and coalitional strategy-pro of ). A distributed algorithm is develop ed for MC, in which intermediate no des in the tree receive messages, p erform some computation, and send messages to their neighb ors. The metho d, a b ottom-up followed by a top-down traversal of the tree, computes the solution to MC with minimal communication complexity, with exactly two messages sent p er link. In comparison, there is no metho d for the SH mechanism with ecient communication complexity. All solutions are maximal, and require as many messages p er link as in a naive
centralized approach. Hence, communication complexity considerations lead to a strong preference for the MC mechanism, which is not budget-balanced. The study leaves many interesting op en questions; e.g. are all budget-balanced solutions maximal, and what are the game-theoretic prop erties of alternative strategy-pro of minimal mechanisms? The economic literature contains a few notable mo dels of the e ect of limited commu- nication and agent b ounded-rationality in mechanism design, and in systems of distributed decision making and information pro cessing. This work is relevant here, given the fo cus in my dissertation on computational mechanism design and in particular on the costs of complete information revelation. In the theory of teams [MR72 ], Radner and Marschak provide a computational account of the organization of management structures and teams, considering in particular the ef- cient use of information within a decentralized organization. One imp ortant assumption made in the theory of teams is that all agents share a common goal (e.g. pro t), no at- tention is given to the incentives of agents. The goal is to compare the eciency (decision quality) of di erent information structures under the assumption that each structure will b e used optimally. The theory of teams prop oses a two-step metho d to measure the e ec- tiveness of a particular organizational structure: (1) nd the optimal mo de of functioning given a structure and compute the eciency; (2) subtract the costs of op eration. The second step in this metho dology has not b een done b ecause there has traditionally b een no go o d way to assess the cost of communication. One metho d suggested to side-step this problem is to compare the p erformance of di erent communication structures for a xed numb er of messages. The work of Feigenbaum et al. [FPS00 ] certainly starts to integrate communication complexity analysis into mechanism design. Radner [Rad87 ] compares the eciency of four classic mo dels of resource allo cation, and asks which is the minimal sucient structure to compute ecient solutions. Extensions to consider agent incentives are also discussed. Recently, Radner [Rad92 , Rad93 ] has considered a decision-theoretic mo del of a rm, in which managers are mo deled as b ounded-rational decision makers, able to p erform some information pro cessing and communicate. The mo del considers distributed decision problems in which agents must p erform lo cal computation with lo cal information b ecause of b ounded-rationality and limited computation. One useful concept prop osed by Radner is that of a \minimally ecient" network, which is the minimal communication network
