Game Theory Giorgio Fagiolo giorgio.fagiolo@univr.it https://mail.sssup.it/ fagiolo/welcome.html Academic Year 2005-2006 University of Verona
Summary 1. Why Game Theory? 2. Cooperative vs. Noncooperative Games 3. Description of a Game 4. Rationality and Information Structure 5. Simultaneous-Move (SM) vs. Dynamic Games 6. Analysis 7. Examples 8. Problems and Suggested Solutions
Analysis Main Question What outcome should we expect to observe in a game played by fully rational players with perfect recall and common knowledge about the structure of the game? Answer It depends on whether: Agents know about others payoffs Rationality is common knowledge Players conjectures about each other s play must be mutually correct Notation s = {s i, s i } = {s i, (s 1,..., s i 1, s i+1,..., s N )} S = S 1 S 2 S N S i = S 1 S 2 S i 1 S i+1 S N Pure vs. Mixed Strategies
Level-1 Rationality Assume H1 only: Players are rational and know the structure of the game. Definition 1 (Strictly dominant strategy) A strategy s i S i is a strictly dominant strategy for player i in game Γ N if for all s i s i and s i S i π i (s i, s i ) > π i (s i, s i ). Definition 2 (Strictly dominated strategies) A strategy s i S i is strictly dominated (SD) for player i in game Γ N if there exists another strategy s i S i such that for all s i S i π i (s i, s i ) > π i (s i, s i ). A rational player satisfying H1 will: play a strictly dominant strategy if there exists one; not play a strictly dominated strategy. Problem: the outcome of the game is far from being unique It is rare that a strictly dominant strategy exists Many options resist to deletion of strictly dominated strategies
Level-1 Rationality: Examples A Strictly Dominant Strategy in the Prisoner Dilemma Player 1 Player 2 (π 1, π 2 ) DC C DC ( 2, 2) ( 10, 1) C ( 1, 10) ( 5, 5) Deletion of a Strictly Dominated Strategy (D for Player 1) Player 2 (π 1, π 2 ) L R U (+1, 1) ( 1, +1) Player 1 M ( 1, +1) (+1, 1) D ( 2, +5) ( 3, +2) No strict dominance in Matching Pennies 2.0 Player 2 (π 1, π 2 ) H T Player 1 H ( 1, +1) (+1, 1) T (+1, 1) ( 1, +1)
Level-2 Rationality (1/3) Assume that H1, H2 and H3 hold: Players know others payoffs and that all are rational. Players know that others will not play strictly dominated strategies. Deletion of strictly dominated strategies can be iterated. Unique predictions can be sometimes reached. Example Player 1 Player 2 (π 1, π 2 ) DC C DC (0, 2) ( 10, 1) C ( 1, 10) ( 5, 5) C is no longer a dominant strategy for player 1. C is still a dominant strategy for player 2. Player 1 knows that 2 will not play DC and can eliminate it. Now player 1 knows that Player 2 will play C, to which C is a dominant strategy.
Level-2 Rationality (2/3) Iterative deletion of strictly dominated strategies (IDSDS) does not depend on the order of deletion. When N = 2 the set of strategies that resist to IDSDS is the prediction of the game. When N > 2 assumption H1-H3 allows one to delete even more outcomes. Definition 3 (Best Response) In game Γ N a strategy s i S i is a best response for player i to s i if for all s i S i π i (s i, s i ) π i (s i, s i ). A strategy s i is never a best response if there is no s i for which s i is a best response. A strictly dominated strategy is never a best response, but there may exist strategies that are never a best-response which are not strictly dominated. Iterative elimination of strategies that are never a best-response leads to the set of rationalizable strategies, which is generally smaller than the set of strategies that resist IDSDS. Rationalizable strategies are those that one can expect to occur in a game played by rational agents for which H1-H3 hold true.
Level-2 Rationality (3/3) Example Player 2 (π 1, π 2 ) l m n U (5, 3) (0, 4) (3, 5) Player 1 M (4, 0) (5, 5) (4, 0) D (3, 5) (0, 4) (5, 3) Best-Response Strategies U if s 2 = l n if s 1 = U s 1 = M D if if s 2 = m s 2 = n, s 2 = m l if if s 1 = M s 1 = D Thus: No strategy is never a best response. Iterative elimination cannot be applied (Hint: with 2 players a strategy is never a BR it is strictly dominated). All strategies can be rationalized by H1-H3 through a chain of justifications. Example for (U, l): J 1 ={P1 justifies U by the belief that P2 will play l, which can be justified if P1 thinks that P2 believes that P1 plays D}. J 2 ={P1 justifies J 1 by thinking that P2 thinks that P1 believes that P2 plays l}... and so on! Beliefs can be mutually wrong! H1-H3 do not require them to be mutually consistent!
Level-3 Rationality (1/3) Assume that H1-H3 hold and that players are mutually correct in their beliefs Definition 4 (Nash Equilibrium (NE)) A strategy profile (s 1,..., s N ) is a Nash equilibrium for the game Γ N if for every i I and for all s i S i π i (s i, s i ) π i (s i, s i ) i.e. if each actual player s strategy is a best-response. Examples: Player 2 (π 1, π 2 ) l m n U (5, 3) (0, 4) (3, 5) Player 1 M (4, 0) (5,5) (4, 0) D (3, 5) (0, 4) (5, 3) Player 1 Player 2 (π 1, π 2 ) DC C DC ( 2, 2) ( 10, 1) C ( 1, 10) (-5,-5)
Level-3 Rationality (2/3) Three Crucial Questions: Why should players play a NE? Does a NE always exist? When a NE exists, is it always unique?
Level-3 Rationality (2/3) Three Crucial Questions: Why should players play a NE? Does a NE always exist? When a NE exists, is it always unique? Why should players beliefs be mutually correct? Not a consequence of rationality NE as obvious ways to play the game NE as Pre-play commitments NE as Social Conventions No reasons to expect players to play NE are given in the rules of the game Need to complement the theory with something else...
Level-3 Rationality (3/3) Does a NE always exist? No, even in simplest games... Player 2 (π 1, π 2 ) H T Player 1 H ( 1, +1) (+1, 1) T (+1, 1) ( 1, +1) Existence is guaranteed in the space of mixed strategies for finite-strategy games Proof: See MWG p.250-253 (Hemicontinuous Corr, Kakutani Fixed Point Theorem)
Level-3 Rationality (3/3) Does a NE always exist? No, even in simplest games... Player 2 (π 1, π 2 ) H T Player 1 H ( 1, +1) (+1, 1) T (+1, 1) ( 1, +1) Existence is guaranteed in the space of mixed strategies for finite-strategy games Proof: See MWG p.250-253 (Hemicontinuous Corr, Kakutani Fixed Point Theorem) When a NE exists, is it always unique? Player 2 (π 1, π 2 ) A B Player 1 A (1,1) (0, 0) B (0, 0) (2,2) Uniqueness and efficiency are not guaranteed When multiple NE arise, no selection principle is given
Summary 1. Why Game Theory? 2. Cooperative vs. Noncooperative Games 3. Description of a Game 4. Rationality and Information Structure 5. Simultaneous-Move (SM) vs. Dynamic Games 6. Analysis 7. Examples 8. Problems and Suggested Solutions
Some Examples Focus on 2-player 2 2 symmetric games (π i = π all i I) Adding (heterogeneous) players and strategies only complicates the framework Games become more difficult to solve No new intuitions: problems always are existence and uniqueness +1 1 +1 a b 1 c d +1 1 +1 1 0 1 α β where: a d, a > b and α = c b a b, β = d b a b so that α R and β 1
Case I: Prisoners Dilemma Games Call +1= Cooperate and 1= Defect : +1 1 +1 a b 1 c d +1 1 +1 1 0 1 α β P D : c > a > d > b. P D : 0 < β < 1 and α > 1 +1 is strictly dominated by 1 ( 1, 1) is the unique (inefficient) Nash
Case II: Coordination Games (CG) We have that: +1 1 +1 a b 1 c d +1 1 +1 1 0 1 α β CG : a > c and d > b CG : 0 < β 1 and α < 1 There are two NE: (+1, +1) and ( 1, 1) β < 1 : (+1, +1) is Pareto-efficient, ( 1, 1) is Pareto-inferior β = 1 : (+1, +1) and ( 1, 1) are Pareto-equivalent A Pure-Coordination Game arises when α = 0.
Case IIa: CG with Strategic Complementarities A 2 person, 2 2 symmetric game is a game with strategic complementarities if the expected payoff from playing +1 (resp. 1) is increasing in the probability that the opponent is playing +1 (resp. 1). Applications: Technological adoption, technological spillovers, etc.. A 2 person, 2 2 symmetric game is a game with strategic complementarities if it is a coordination game and in addition: SC : a > b, d > c SC games are therefore characterized by: SC : 1 α 1, 0 β 1 and 1 α β < 0 +1 1 +1 1.0 0.0 1 0.5 0.8
Case IIb: Stag-Hunt Coordination Games A 2 person, 2 2 symmetric game is called a stag-hunt game if (i) (+1, +1) is Pareto dominant (a > d); and (ii) c > d. SH : 1 α 1, 0 β 1 and 0 α β < 1 +1 1 +1 1.0 0.0 1 0.5 0.3 Applications: Pre-Play Commitments. Suppose that player I commits in a pre-play talk to play the efficient strategy +1. Would he be credible? No. Why? Because player I cannot credibly communicate this intention to player II as it is always in player I s interest to convince II to play +1. Indeed, if player I is cheating, he is going to get always a larger payoff if player II will play +1, as c > d. Hence, by convincing II to play +1, player I will always get a gain and the pre-play commitment is not credible. That is why pre-play communication does not ensure efficiency in a SH game. Conversely, in SCG, pre-play commitment is credible and can ensure efficiency.
Case III: Hawk-Dove Games A 2 person, 2 2 symmetric game is called a hawk-dove game if: HD : c > a > b > d HD : α 1 and β 0 Interpretation Strategy +1 is dove and strategy 1 is hawk. There is a common resource of 2 units. If two +1 meet, they share equally and get a payoff of 1 each. If a +1 and a 1 meet, then +1 gets 0 and 1 gets all 2 units. If two 1 meet, then not only they destroy the resource, but they get a negative payoff of 1 each. +1 1 +1 1 0 1 2 1 There are two NE: (+1, 1) and ( 1, +1)
Case IV: Efficient Dominant Strategy Games PD Games have an inefficient dominant strategy. dominant strategy: EDS games have instead an efficient EDS : a > c and b > d EDS : α < 1 and β < 0 +1 1 +1 1 0 1 α β 1 is strictly dominated by +1 (+1, +1) is the unique efficient NE
Classification of 2-person 2 2 Symmetric Games
Classification of 2-person 2 2 Symmetric Games
Summary 1. Why Game Theory? 2. Cooperative vs. Noncooperative Games 3. Description of a Game 4. Rationality and Information Structure 5. Simultaneous-Move (SM) vs. Dynamic Games 6. Analysis 7. Examples 8. Problems and Suggested Solutions
I. Incomplete Information (1/2) What happens if information is incomplete (nature moves first) and therefore imperfect (info sets are not singletons)? Example: Harsanyi Setup 1. Nature chooses a type for each player (θ 1,..., θ N ) Θ = Θ 1 Θ N 2. Joint probability distribution F (θ 1,..., θ N ) common knowledge 3. Each player can only observe θ i Θ i 4. Payoffs to player i are: π i (s i, s i ; θ i ) 5. A pure strategy for i is a decision rule s i (θ i ) 6. The set of pure strategies for player i is the set R i of all possible decision rules (i.e. functions s i (θ i )) 7. Player i s expected payoff is given by: π i (s 1 ( ),..., s N ( )) = E θ [π i (s 1 (θ 1 ),..., s N (θ N ); θ i )]
I. Incomplete Information (2/2) Bayesian Nash Equilibrium Definition 5 A Bayesian Nash Equilibrium (BNE) for the game [I, {S i }, {π i ( )}, Θ, F ( )] is a profile of decision rules (s 1 ( ),..., s N ( )) that is a NE for the game [I, {R i }, { π i ( )}]. Amount of information and computational abilities required is huge! In a BNE each player must play a BR to the conditional distribution of his opponents strategies for each type θ i he can have! Each player is actually split in a (possibly infinite) number of identities, each one associated to an element of Θ i It is like the game were populated by a (possibly infinite) number of players Only very simple games can be handled Multiple players? Multiple Nature Moves? Commitment to full rationality and complete-graph interactions become too stringent
II. Equilibrium Selection (1/3) The theory is completely silent on: Why players should play a NE What happens when a NE does not exist What happens when more than one NE do exist (e.g., efficiency issues) Complementing NE theory with equilibrium refinements Additional criteria that (may) help in solving the equilibrium selection issue Similar to tâtonnement in general equilibrium theory So many refinement theories that the issue shifted from equilibrium selection to equilibrium refinements selection... Each refinement theory can be ad hoc justified Example: Trembling Hand Perfection (MWG, 8.F) Consider the mixed-strategy game Γ N = [I, (S i ), π i ] The mixed-strategy set (S i ) means that players choose a probability distribution over S i, i.e. play a strategy σ i in the K-dim simplex A pure strategy is a vertex of the simplex The boundary of the simplex means playing some strategy with zero probability
II. Equilibrium Selection (2/3) Example: Cont d Problem: Some pure-strategy Nash equilibria are the result of an excess of rationality, i.e. they did not occurred if the players knew how to move slightly away from them NE with weakly-dominated strategies A B A 4 3 B 0 3 Question: What if we force players to play every pure-strategy with a small but positive probability (make mistakes)? Players now must choose a strategy in the interior of the simplex More formally: Define ɛ i (s i ) for all s i S i and i I and allow players to choose mixed strategies s.t. the probability that player i plays the pure-strategy s i is larger that ɛ i (s i ). Define a perturbed-game Γ N,ɛ as the original game Γ N when a particular choice of lower-bounds ɛ i (s i ) for all s i S i and i I is made. A mixed-strategy NE σ of the original game will be a Trembling-Hand Perfect (THP) NE if there exists a sequence of perturbed games that converges to the original game (as mistake sizes go to zero) whose associated NE stay arbitrarily close to σ.
II. Equilibrium Selection (3/3) A THP NE always exists if the game admits a finite number of pure strategies Main Result: If σ is a THP Nash equilibrium, then it does not involve playing weaklydominated strategies. If we decide not to accept equilibria that involve weakly-dominated strategies, and we are dealing with games with a finite number of pure strategies, we are sure that at least a THP NE does exist. Problems: Some important games have an infinite number of pure-strategies (continuous): Bertrand oligopoly games Difficult to find out THP NE if the game becomes more complicated (multiple players, incomplete information, etc.) Again: Commitment to full rationality and complete-graph interactions become too stringent
Concluding Remarks (1/2) Theory for simultaneous-move games very useful to answer very (too?) ( low fat modeling ) simple questions Restrictions on analytical treatment imposed by rationality requirements and interaction structure (all interact with everyone else) often become too stringent Games become very easily untractable and/or generate void implications when Many heterogeneous players Information is incomplete and/or imperfect Interaction structure not a complete graph Moves are not simultaneous: Dynamic games and anything can happen kind of results Need to go beyond full rationality paradigm, representative-individual philosophy and peculiar interaction structures Relevant literature: Evolutionary-game theory and beyond
An alternative class of models Agents: i I = 1, 2,..., N Actions: a i A i = {a i1,..., a ik } Time: t = 0, 1, 2,... Dynamics: Concluding Remarks (2/2) At each t (some or all) agents play a game Γ with players in V i Interaction sets V i I define the interaction structure Agents are boundedly-rational and adaptive: they form expectations by observing actions played in the past by their opponents Time-t payoffs to agent i are give by some function w t (a; a t 1 j, j V i ) where a A i Players update their current action at each t e.g. by choosing their myopic BR to observed configuration a t i arg max a A i w t (a; a t 1 j, j V i ) Looking for absorbing states or statistical equilibria of the (Markov) process governing the evolution of an action configuration (or some statistics thereof) as t a t = (a t 1,..., a t N )