Game Theory II: Maximin, Equilibrium, and Refinements Adam Brandenburger J.P. Valles Professor, NYU Stern School of Business Distinguished Professor, NYU Polytechnic School of Engineering Member, NYU Institute for the Interdisciplinary Institute of Decision Making Associated Faculty Member, Center for Data Science
What is the mathematcal definiton of a game? What is the definiton of a strategy? How can one analyze choice under complete ignorance? Von Neumann, J., Zur Theorie der GesellschaJsspiele, Mathema'sche Annalen, 100, 1928, 295-320. 7/9/15 13:31 (Bargman, S., English translaton: On the Theory of Games of Strategy, in Tucker, A., and R.D. Luce (eds.), 2 Contribu'ons to the Theory of Games, Volume IV, Princeton University Press, 1955. 13-42.)
Von Neumann s maximin (best worst case, or best guarantee) decision rule: Ann chooses a strategy to solve Bob chooses a strategy to solve max!! min!!!! (!!,!! )! max!! min!!!! (!!,!! )! Players are allowed to choose mixed not just pure strategies, and we calculate expected payoffs Choice of a mixed strategy can raise a player s maximin payoff (if the other player minimizes relatve to the mixture and not the realizaton) NoTce that in von Neumann theory, players avoid a predictve approach 7/9/15 13:31 Von Neumann op.cit. 3
Ann is about to choose a maximin strategy Before she does, she asks herself: Suppose Bob is choosing a maximin strategy. What, then, is my best choice of strategy? If the answer to this queston is not a maximin strategy for Ann, then she might queston the consistency of the maximin rule Von Neumann s famous Minimax Theorem implies that for two- player zero- sum games, i.e., games where a maximin strategy remains optmal for Ann (and likewise with Ann and Bob interchanged)!!!!,!! +!!!!,!! = 0! 7/9/15 13:31 Von Neumann op.cit. 4
The von Neumann- Morgenstern theory of n- player general- sum games: Each coali4on (subset) of players jointly chooses strategies, under the maximin assumpton about how the complementary set of players chooses This yields a superadditve characteris4c func4on that associates with each subset S of the player set N, a real number (the value jointly created by S) CooperaTve game theory - - - via soluton concepts such as the Core - - - analyzes the creaton and division of value in this model Theorem (von Neumann- Morgenstern): Every superadditve characteristc functon can be built from a matrix game and maximin behavior Non- cooperatve branch: Procedural game theory Assumes individualism CooperaTve branch: Combinatorial game theory Assumes coordinaton 7/9/15 13:31 Von Neumann op.cit. 5
http://upload.wikimedia.org/wikipedia/commons/0/04/nassau_hall_princeton.jpg http://www.pnas.org/site/classics/classics5.xhtml http://www.awesomestories.com/asset/view/john-nash-photo-as-a-young-man John von Neumann: Each player must choose his strategy in complete ignorance 7/9/15 13:31 John Nash: [A] ra4onal predic4on should be unique 6
A Nash equilibrium is a mixed- strategy profile (σ 1,, σ n ) such that, for each player i, the mixed strategy σ i maximizes player i s expected payoff, given that each other player j i chooses the mixed strategy σ j Theorem (Nash): Every game with finitely many players, each with a finite pure strategy set, possesses at least one Nash equilibrium Nash, J., Non- CooperaTve Games, doctoral dissertaton, Princeton University, 1950; available at 7/9/15 13:31 hkp://rbsc.princeton.edu/topics/nash- john- 1928-2015; Nash, J., Non- CooperaTve Games, Annals of 7 Mathema'cs, 54, 1951, 286-295
Ann Bob Le? Right le? right 1 1 2 2 0 0 There is a Nash equilibrium in which Ann chooses R Does Nash equilibrium involve too likle ratonality on the part of the players? 7/9/15 13:31 8
The CenTpede Game: Ann Bob In In In In In Out Out Out Out Out Out In $19 $22 $2 $1 $1 $4 $4 $3 $3 $6 $17 $20 $20 $19 In any (even mixed) Nash equilibrium of the CenTpede game, Ann chooses Out immediately Does Nash equilibrium involve too much correctness on the part of the players? Rosenthal, R., Games of Perfect InformaTon, Predatory Pricing, and the Chain Store, Journal of Economic 7/9/15 13:31 Theory, 1980, 25, 92-100; McKelvey, R., and T. Palfrey, An Experimental Study of the CenTpede Game, 9 1992, Econometrica, 60, 803-836; hkp://commons.wikimedia.org/wiki/file:stck_figure.jpg
7/9/15 13:41 Flood, M., Some Experimental Games, Management Science, 5, 1958, 5-26 10
The refinement program: Backward inducton (von Neumann- Morgenstern 1944) Subgame- perfect equilibrium (Selten 1965) Trembling- hand perfect equilibrium (Selten 1975) Proper equilibrium (Myerson 1978) SequenTal equilibrium (Kreps and Wilson 1982) Forward inducton (Kohlberg and Mertens 1986) Stable equilibrium (Mertens 1989) How can we study ratonality in games without assuming equilibrium? Later Selten, R., SpieltheoreTsche Behandlung eines Oligopolmodells mit Nachfragentragheit, Zeitschri? fur die gesamte Staatswissenscha?, 12, 1965, 201-324 Selten, R., ReexaminaTon of the Perfectness Concept of Equilibrium Points in Extensive Games, Interna'onal Journal of Game Theory, 4, 1975, 25-55 Myerson, R., Refinements of the Nash equilibrium Concept, Interna'onal Journal of Game Theory, 7, 1978, 73-80 Kreps, D., and R. Wilson, SequenTal Equilibria, Econometrica, 50, 1982, 863-894 Kohlberg, E., and J- F. Mertens, On the Strategic Stability of Equilibria. Econometrica, 54, 1986, 1003-1037 Mertens, J- F., ``Stable Equilibria - - - A ReformulaTon, Mathema'cs of Opera'ons Research, 14, 1989, 575-625 7/9/15 13:31 11
Forward InducTon: 2 * Out Le? Ann Right Bob le? right le? right 3 3 0 0 0 0 1 1 There is a Nash equilibrium in which Ann chooses Out But if Bob gets to move, he knows Ann has foregone a payoff of 2 7/9/15 13:31 Kohlberg and Mertens op.cit. 12