Eric Maskin

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Nash Equilibrium and Mechanism Design
∗ 
 
 
 
 
 
 
 
Eric Maskin 
 
Institute for Advanced Study 
 
and 
 
Princeton University 
 
November 2008 
 
 
 
 
 
 
 
                                                 
∗ This short paper was prepared for a conference at Princeton University June 13-14, 2008 in celebration of 
John Nash’s 80
th birthday. I thank the NSF for research support.   1
 A  Nash equilibrium (called an “equilibrium point” by John Nash himself; see 
Nash 1950) of a game occurs when each player chooses a strategy from which unilateral 
deviations do not pay.  The concept of Nash equilibrium is far and away Nash’s most 
important legacy to economics and the other behavioral sciences.  This is because it 
remains the central solution concept—i.e., prediction of behavior—in applications of 
game theory to these fields.  As I shall review below, Nash equilibrium has some 
important shortcomings, both theoretical and practical.  I will argue, however, that these 
drawbacks are far less troublesome in problems of mechanism design than in many other 
applications of game theory. 
1. Solution Concepts 
Game-theoretic solution concepts divide into those that are noncooperative—
where the basic unit of analysis is the individual player—and those that are cooperative, 
where the focus is on coalitions of players.  John von Neumann and Oskar Morgenstern 
themselves viewed the cooperative part of game theory as more important, and their 
seminal treatise, von Neumann and Morgenstern (1944), devoted fully three quarters of 
its space to cooperative matters.  This pattern was followed by leading game theory 
textbooks, such as Luce and Raiffa (1957) and Owen (1970) up until the mid-1980s.  
Today, by contrast, Nash equilibrium—the noncooperative concept par excellence—
dominates the standard textbooks, and such leading texts as Fudenberg and Tirole (1991), 
Myerson (1991), and Osborne and Rubinstein (1994) give short shrift (or no shrift at all) 
to the cooperative side.   2
  I believe that there are three (related) reasons for the historical shift from 
cooperative to noncooperative theory
1:(a) most cooperative theory ignores externalities, 
the possibility that a coalition can be affected by the actions of those not in the coalition; 
(b) it assumes that a Pareto efficient outcome will be reached; and (c) it supposes that the 
grand coalition (the coalition of all players) will form.  Point (a) corresponds to the fact 
that typically games in cooperative theory are represented in characteristic function form, 
according to which the payoffs that any particular coalition can achieve are independent 
of which other coalitions form.  Point (b) is embodied in the efficiency axioms typically 
imposed by cooperative solution concepts, requiring outcomes to be Pareto efficient, and 
(c) is implied by efficiency in superadditive games
2 (which are the games usually studied 
in the literature).  These features of cooperative theory are problematic because most 
applications of game theory to economics involve settings in which externalities are 
important, Pareto inefficiency arises, and the grand coalition does not form.  To see this, 
one need look no further than the classic game-theoretic model, Cournot duopoly. 
  Nash himself proposed a unification of noncooperative and cooperative theory 
that has come to be known as the Nash Program (Nash 1951).  Yet, while there has been 
important work following this idea up in the theoretical literature (see Serrano 2005 for a 
recent survey), the Nash Program has not yet had much effect on applications. 
  Nash equilibrium has been successful as a solution concept first because it is 
logically coherent.  Specifically, it is the only concept that is consistent both with (1) 
expected payoff-maximization by players (rational behavior) and (2) correct forecasts by 
players about what others will do (rational expectations).  Moreover, it has proved to 
                                                 
1 For a fuller exposition of these points, see Maskin (2003). 
2 A game is superadditive if the union of two disjoint coalitions can obtain at least the sum of the payoffs of 
the two separate coalitions.   3
make good predictions of behavior both in experimental and field settings, at least when 
subjects have acquired sufficient experience playing the game in question. 
2. Drawbacks 
  Nevertheless, Nash equilibrium has several important shortcomings.  First, many 
games have multiple equilibria, and players may not be clear about which one to focus on. 
If the players can communicate with each other before the game is played, they may be 
able to select an equilibrium through negotiation (that is why a Nash equilibrium is 
sometimes referred to as a “self-enforcing agreement”).  But negotiation does not always 
suffice to resolve multiplicity.  Consider, for example, the game of Table 1 (borrowed 
from Aumann 1990), in which (U, L) and (D, R) are both equilibria (there is also a mixed 
strategy equilibrium).  Players may attempt to negotiate the outcome (U, L), which Pareto 
dominates the other equilibrium.  Thus, player 1 will announce that she plans to play U 
and player 2 that he plans to play L.  Notice, however, that these professions may not be 
credible.  In particular, regardless of what she does herself, player 1 is better off if player 
2 takes action L.  But player 2 will play L only if he thinks there is a sufficiently high 
 
                                   
 
Table 1   4
probability that 1 will choose U.  Thus, player 1 has the incentive to say that she intends 
to play U regardless of whether that is actually true.  Moreover, because U is a risky 
strategy for 1 (she could get a payoff as low as 0 with U, whereas with D her lowest 
possible payoff is 7), she might well play D if she is not very confident that 2 will play L.  
In other words, her announcement that she will play U is not really believable, and 
neither is the announcement L by player 2.  Negotiation between the two players may, 
therefore, not accomplish much. 
  Of course, even without communication, multiple equilibria do not always cause a 
problem.  Consider, for example, the game of Table 2.  There are two equilibria: 
                           
Table 2 
 
(U, L) and (D, R) (plus a mixed strategy).  However, (U, L) stands out as obviously 
superior; it is focal in the sense of Schelling (1960).  Unfortunately, not all games have 
one particular equilibrium toward which players will naturally gravitate.  For instance, in 
the game of Table 3—the classic Battle of the Sexes—the equilibria   5
                                 
 
Table 3 
 
(U, L) and (D, R) are exactly symmetric, and so there is no obvious criterion that would 
direct players to one equilibrium rather than the other. 
  Even when Nash equilibrium is unique, rationality on the part of the players does 
not by itself guarantee that equilibrium will be reached, as a player’s theory about what 
the others will do may be incorrect.  Look at the game in Table 4. 
 
Table 4 
   6
There is a unique Nash equilibrium ( ) 22 , ab.  However, any other outcome is 
rationalizable (see Bernheim 1984 and Pearce 1984) in the sense of being consistent with 
the players’ common knowledge of their rationality.  For example, it is optimal for player 
1 to play  1 a  if she anticipates that player 2 will play  3 b .  And this anticipation is justified 
if player 1 thinks 2 has reason to believe that 1 will play  3 a , etc.. 
3. Mechanism Design 
  Although all these problems with Nash equilibrium are important, they are 
typically much less severe than usual when the game at hand is the outcome of 
mechanism design.  The theory of mechanism design is the “engineering” part of 
economic theory.  One starts with a particular goal or objective and then enquires if and 
how a mechanism—that is, a game—could be designed that attains that goal in 
equilibrium (in which case the game is said to implement the goal).  In other words, the 
game is chosen not given. 
  There are three major analytical advantages that accrue from a game being chosen 
by a mechanism designer rather than simply being given by “nature.”  First, since the 
designer can specify the rules in advance, the players themselves should presumably 
know exactly which game is being played.  Consider, by comparison, the uncertainty that, 
Ford, Chrysler, and General Motors face in their “game” with one another: the timing, the 
possible moves, and the possible payoffs are all quite unclear. 
  Second, in mechanism design, the analyst observing and studying a games’s 
execution also knows the rules of the game.  And this feature makes experimental and 
empirical field work considerably easier than in the usual case, where—as in the 
automobile industry—we have only highly simplified and very approximate models to go   7
by.  Indeed, one complaint about experimental economics is that lessons learned in the 
lab are at times difficult to apply to the field because the respective games in the two 
settings are not guaranteed to be the same.  But in mechanism design, the games can be 
constructed to be the same, and so the standard objection no longer applies. 
  Finally, in mechanism design, the games themselves can be chosen to have 
attractive properties.  For example, in some standard settings, implementing games can 
be constructed so that they have a unique Nash equilibrium (see Maskin and Sjöström 
2003).  In fact, one can sometimes ensure that there are no rationalizable outcomes other 
than this unique equilibrium.  Specifically, Abreu and Matsushima (1992) show that this 
is possible quite generally if one doesn’t require that goals be achieved exactly but only 
approximately. 
  It is even sometimes possible to construct implementing mechanisms with 
equilibria that are considerably stronger than ordinary Nash equilibrium.  For instance, in 
auction settings with private values in which buyers have quasilinear utility for the good 
being sold, one can attain the goal of efficiency by means of a Vickrey (second-price 
auction), in which bidding one’s actual valuation is a dominant strategy (i.e., regardless 
of what others do, bidding one’s valuation is optimal).  Moreover, in some interdependent 
values settings, a generalization of a Vickrey auction achieves much the same thing (but 
with ex post equilibrium rather than dominant-strategy equilibrium); see Crémer and 
McLean 1988, Dasgupta-Maskin 2000, and Bergemann and Morris 2008. 
  Thus, from several perspectives, Nash equilibrium is a much less problematic 
solution as used in mechanism design than in many other areas of economics and political   8
science.  Indeed, mechanism design provides the circumstances perhaps most favorable 
for Nash equilibrium being a good predictor of human behavior in strategic settings.   9
References 
Abreu, D. and H. Matsushima (1992), “Virtual Implementation in Iteratively 
Undominated Information: Complete Information,” Econometrica, 60, 993-1108. 
 
Aumann, R.J. (1990), “Nash Equilibria are not Self-Enforcing,” in J-J Gabsewicz, J-F 
Richard, and L. Wolsey (eds), Economic Decision-Making: Games, Econometrics, 
and Optimisation, Amsterdam: North-Holland, 201-206. 
 
Bergemann, D. and S. Morris (2008), “Ex Post Implementation,” Games and Economic 
Behavior, 63, 527-566. 
 
Bernheim, B.D. (1984), “Rationalizable Strategic Behavior,” Econometrica, 52, 1007-
1028. 
 
Crémer, J. and R. McLean (1988), “Full Extraction of the Surplus in Bayesian and 
Dominant Strategy Auctions,” Econometrica, 56, 1247-1257. 
 
Dasgupta, P. and E. Maskin (2000),“Efficient Auctions,” Quarterly Journal of 
Economics, Vol. CXV, May 2000, 341-388. 
 
Fudenberg, D. and J. Tirole (1991), Game Theory, Cambridge: MIT Press. 
 
Luce, R.D. and H. Raiffa (1957), Games and Decisions, New York: J. Wiley and Sons. 
 
Maskin, E. (2003), “Bargaining, Coalitions, and Externalities,” Presidential address, 
Econometric Society. 
 
Maskin, E. and T. Sjöström (2002), “Implementation Theory,” in K. Arrow, A. Sen, and 
K. Suzumura (eds.), Handbook of Social Choice Theory Vol. I, Amsterdam: North 
Holland, pp. 237-288.  
 
Myerson, R.B. (1991), Game Theory: Analysis of Conflict, Harvard University Press. 
 
Nash, J.F. (1950), “Equilibrium Points in n-Person Games,” Proceedings of the National 
Academy of Sciences, 36, 48-49. 
 
Nash, J.F. (1951), “Noncooperative Games,” Annals of Mathematics, 54, 289-295. 
 
Osborne, M. and A. Rubinstein (1994), A Course in Game Theory, Cambridge: MIT 
Press.  
 
Owen, G. (1970), Game Theory, New York: Academic Press. 
 
Pearce, D. (1984), “Rationalizable Strategic Behavior and the Problem of Perfection,” 
Econometrica, 52, 1029-1050.   10
 
Serrano, R. (2005), “Fifty Years of the Nash Program, 1953-2003,” Investigaciones 
Ecónomicas, 29, 219-258. 
 
von Neumann, J. and O. Morgenstern (1944), Theory of Games and Economic Behavior, 
Princeton University Press. 
 

Nash Equilibrium and Mechanism Design

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