We can now present 28 See Section 11 in Khan-Sun , which provides an analog of the ex-post Walrasian equilibria con- sidered in Section 3 of Sun The quote is taken from Section Let G U be a large game with idiosyncratic uncertainty.
A two-line proof of this proposition is furnished in Khan-Sun The basic idea is straightforward. And we can now finish the proof by appealing to the following proposition which is Theorem 2 of Sun and to the ELLN as stated in Proposition 5. Proposition 7. And so rather than the proof, it is the interpretation of Proposition 6 that is of interest. The context is one of exogenous uncertainty whereby the individual payoffs, as well as the individual randomized strategies, are independent, and the proposition rigorously develops the intuition that once uncertainty is resolved, a player has no incentive to depart ex-post from her optimal strategy taken in the ex-ante game when she finds herself in the realized ex-post game.
In this connection, but in the context of a large but finite game, Kalai writes: A particular modeling difficulty of noncooperative game theory is the sensitivity of Nash equilibrium to the rules of the game, e.
Since such details are often not available to the modeler or even to the players of the game, equilibrium prediction may be unreliable. For this purpose, we define a Nash equilibrium of a game to be extensively robust31 if it remains a Nash equilibrium in all extensive versions of the simultaneous-move game.
Extensive robustness means in particular that an equilibrium must be ex-post Nash. Even with perfect hindsight knowledge of the types and selected actions of all of his opponents, no player regrets, or has an incentive to revise, his own selected action.
But the point can be sharpened still if rather than work with a large game with idiosyncratic uncertainty G U , one works instead with a deterministic large game as in Definition 1.
Aumann also works with a process and with mixed and behavior strategies, but bypasses measurability considerations arising out of the independence issue. In other words, this is to ask, in the terminology adopted by Kalai , whether a mixed strategy equilibrium has an ex-post purification. Since the question is being posed in a deterministic large game, an affirmative answer is even easier to obtain than in the situation considered in Proposition 6 above.
The definition of the ex-post property of a mixed strategy profile in a large game is provided as follows. Definition 8. Since a mixed strategy profile can now be rigorously defined in a framework with strategic uncertainty, the definition simply means that it is ex-post Nash if no player has any incentive to unilaterally change her selected action after the realized state, the realized action distri- bution being induced by the selected actions of all other players in the given state.
This observation leads us to the following result for a large game. Theorem 2. Now, suppose that a mixed strategy profile g has the ex-post Nash property.
The intuition behind the proof above is simple. This proves that a MSE must possess the ex-post Nash property. And because the argument can be carried out in reverse, we can also establish that a mixed strategy profile with the ex-post Nash property must be an equilibrium itself. Without belaboring the point, the observation needs to be made that the above result is a complete resolution to the dilemma, identified in Section 2, pertaining to the simultaneous requirement of independence and joint measurability in the modeling of a mixed strategy profile.
It is precisely to bypass this dilemma that Kalai works with an increasing sequence of large but finite games and emphasizes an approximate ex-post Nash notion and an equicon- tinuity property of payoffs that plays no role in the result presented here. However, his discussion of the property itself is illuminating. The ex-post Nash theorem This is the reason for the turn to approximations. In a slightly different setting 33 See, for example, Kalai , Lemma 6.
One may also mention here that in the other direction, as shown in Cartwright- Wooders , Example 2 , a strategy profile that has the approximate ex-post property may not necessarily be an equilibrium, even an approximate one. Consider a one-shot game of complete information modified from a game of regime change by Angeltos et al. The former supports a rich Fubini extension, whereas the Lebesgue unit interval does not.
It is straightforward to see that G is a game that fits Definition 1. And it is easy to check that h is a RSED. The continuum of agents are formalized as a saturated probability space.
Since hyperfinite Loeb counting probability spaces are saturated, these 35 We return to this issue of asymptotic implementation in the next section.
Thus, one can implement this methodology to show that the exact limiting results presented as Theo- rems 1 and 2, as well as the result in Section 11 of Khan-Sun on a large game, have asymptotic analogs for games with a large but finite number of players.
In this section, we present an asymptotic version of Theorem 2, one in which the independence condition across players is only satisfied in an approximate sense, and thereby allows the presentation of a more general asymptotic result.
In some sense, its simplicity and analytical sophistication is also its expository failing. One wants to proceed more slowly so as to unravel the proof and delineate how and why the ELLN and the Fubini property, as well as the continuity assumption on payoffs, are intimately involved in it. Such a discussion,37 heuristic and in- tuitive to be sure, would serve as a natural bridge between it and the large but finite result presented as Proposition 8 below.
This will enable us not only to unravel the result pre- sented as Theorem 2, but also bring out why the naive argument needs supplementation by a rigorous one.
Since the action set is finite, the payoff of an arbitrary player i can now be explicitly written out in terms of relative frequencies with which each action is being played; i. The reader may compare the corresponding one with Kalai We also note here that in this attempt, we take our lead and inspiration from Aumann , Section 8.
This allows the conclusion that the original mixed strategy equilibirum satisfies the ex-post Nash property approximately, 7 being the approximate version of It is important for the reader to appreciate why such a conclusion cannot be had, and where the failure rests.
In turning to this, we need the basic notion of a tight sequence of mappings. We need such an idea of a tight sequence of mappings in order to put some control on the extent to which the characteristics are allowed to vary. Let G n be a game, which is to say, a measurable function from I n to UA. We now develop the notion of asymptotic Nash equilibria in mixed strategies. Definition 9. But this requires elaboration: a rather long and tedious management of different epsilons.
In our setting, fi differs from gi by one point and is thus essentially the same measurable function. Here we simply note that with this result, we obtain the asymptotic ex-post Nash property for large finite games with asymptotically approximate versions of independence and and of a Nash equi- librium.
The result is considerably more general than the case for large finite games with exact versions of independence and and of a Nash equilibrium. After a much-needed and overdue clarification of the measurability problem, we furnish a complete resolution of two long-standing41 open problems: Theorem 1 shows the equivalence between a MSE and a RSED, and Theorem 2 relies on it to exhibit the ex-post Nash property of randomized strategy equilibria.
More generally, our application of the ELLN brings out the ease with which one can perform operations upon a continuum of independent random variables and provides a rigorous measure-theoretic micro-foundation that can be used to model other macroeconomic and microeconomic scenarios.
In terms of future work and direction, note that the notions of a MSE and a RSED admit direct translation to a more elaborate model of a large game, one in which agent-names and agent-traits are disentangled, as considered in Khan et al.
We leave it to the interested reader to check that Theorems 1 and 2 are still valid for such a model, and indeed virtually 41 See Footnote 30 and the text that it footnotes. This is thereby a movement from a large game of complete information to one with a multi-stage incomplete information game; see Morris-Shin , Angeltos et al. This means that the process g is essentially pairwise independent. Angeletos, C. Hellwig and A. Pavan, Dynamic global games of regime change: learning, multiplicity and the timing of attacks, Econometrica 75, — Aumann, On choosing a function at random, in: F.
Wright, ed. New York: Academic Press. Mixed and behavior strategies in infinite extensive games, in: M. Dresher, L. Shapley, and A. Tucker, eds. Princeton: Princeton University Press. Subjectivity and correlation in randomized strategies, J.
Values of markets with a continuum of traders, Econometrica 43, — Aumann and J. Dreze, Rational expectations in games, Amer. Cartwright and M. Wooders, On purification of equilibrium in Bayesian games and expost Nash equilibrium, Int. Game Theory 38, — Cremer and R.
McLean, Optimal selling strategies under uncertainty when demands are interdependent, Econometrica 53, — Dudley, Real Analysis and Probability. New York: Chapman and Hall. Doob, Stochastic processes depending on a continuous parameter, Trans.
Stochastic Processes. New York: Wiley. Dvoretsky, A. Wald and J. Wolfowitz, Relation among certain ranges of vector measures, Pac. Feldman and C. Gilles, An expository note on individual risk without aggregate uncer- tainty, J. Theory 35, 26— Judd, The law of large numbers with a continuum of i. Theory 35, 19— Kalai, Large robust games, Econometrica 72, — Keisler and Y. Sun, Why saturated probability spaces are necessary, Adv. Khan, K. Rath and Y.
The Dvoretzky-Wald-Wolfowitz theorem and purifi- cation in atomless finite-action games, Int. Game Theory 34, 91— Rath, Y. Sun and H. Yu, Large games with a bio-social typology, J. Theory, , — Khan and Y.
Non-cooperative games with many players, in: R. Aumann and S. Hart, eds. Amsterdam: Elsevier Science. Kuhn, Extensive games and the problem of information, Ann. Loeb and M. Wolff, eds. Nonstandard Analysis for the Working Mathematician. Dor- drecht: Kluwer Academic Publishers. Milgrom and R. Weber, Distributional strategies for games with incomplete information, Math. Morris and H. Shin, Global games: theory and applications, in: M. Dewatripont, L. Hansen and S. Turnovsky, eds.
Save to Library Save. Create Alert Alert. Share This Paper. Background Citations. Methods Citations. Results Citations. Topics from this paper. Game theory. Citation Type. Has PDF. Publication Type. More Filters. Note: This is a only a draft version, so there could be flaws. If you find any errors, please do send email to hari csa. A more thorough version would be available soon in this space. View 1 excerpt. Game Theory Lecture Notes by Chapter Highly Influenced.
0コメント