# Equilibrium computation: Part 1

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1 Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

2 Outline 1 Models and solution concepts Mechanisms in strategic form Solution concepts 2 Non equilibrium solution concept computation Finding dominated actions Finding never best response actions 3 Computing a Nash equilibrium with strategic form games Matrix games Bimatrix games Polymatrix games 4 Computing correlation based equilibria with strategic form games Computing a correlated equilibrium Computing a leader follower equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

3 Game model Definition A game is formally defined by a pair: Mechanism M, defining the rules of the game Strategiesσ, defining the behavior of each agent in the game Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

4 Game model Definition A game is formally defined by a pair: Mechanism M, defining the rules of the game Strategiesσ, defining the behavior of each agent in the game Mechanisms There are three main classes of mechanisms: Strategic form mechanisms: agents play without observing the actions undertaken by the opponents (simultaneous games) Extensive form mechanisms: there is a sequential tree based structure according which an agent can observe some opponents actions Stochastic form mechanisms: there is a sequential graph based structure according which an agent can observe some opponents actions Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

5 Games in strategic form (1) Definition A strategic form mechanism is a tuple M = (N,{A} i N, X, f,{u} i N ) N: set of agents A i : set of actions available to agent i X: set of outcomes f : i N A i X: outcome function U i : X R: utility function of agent i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

6 Games in strategic form (2) Example: Rock Paper Scissors N = {agent 1, agent 2} A 1 = A 2 = {R, P, S} X = {win1, win2, tie} f(r, S) = f(p, R) = f(s, P) = win 1, f(s, R) = f(r, P) = f(p, S) = win2, tie otherwise U i (wini) = 1, U i (win i) = 1, U i (tie) = 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

7 Games in strategic form (2) Example: Rock Paper Scissors N = {agent 1, agent 2} A 1 = A 2 = {R, P, S} X = {win1, win2, tie} f(r, S) = f(p, R) = f(s, P) = win 1, f(s, R) = f(r, P) = f(p, S) = win2, tie otherwise U i (wini) = 1, U i (win i) = 1, U i (tie) = 0 Matrix based representation agent 1 agent 2 R P S R 0, 0 1, 1 1, 1 P 1, 1 0, 0 1, 1 S 1, 1 1, 1 0, 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

8 Games in strategic form (3) Example: three player game A 1 = {a, b} A 2 = {L, R} A 3 = {A, B, C} L R a 2, 2, 1 0, 3, 0 b 3, 0, 2 1, 1, 4 A L R a 2, 3, 0 0, 4, 1 b 3, 1, 2 1, 2, 0 B L R a 2, 1, 0 1, 0, 2 b 0, 3, 1 2, 3, 1 C Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

9 Matrix based games Classification Matrix game: the agents utilities can be represented by a unique matrix (this happens with two agent constant sum games: U 1 + U 2 = constant for every entry) Bimatrix game: two agent general sum games Polymatrix game: the utility U i of each agent i can be expressed as a set of matrices U i,j depending only on the actions of agent i and agent j with non polymatrix games, U i has j N A j entries with polymatrix games, U i has A i j N,j i A j entries Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

10 Strategies Definition A strategy σ i of agent i is a probability distribution over the actions A i Call x i,j the probability with which agent i plays action j and x i the vector of x i,j, we need that x i 0 1 T x i = 1 A strategy profileσ is the collection of one strategy per agent, σ = (σ 1,...,σ N ) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

11 Strategies Definition A strategy σ i of agent i is a probability distribution over the actions A i Call x i,j the probability with which agent i plays action j and x i the vector of x i,j, we need that x i 0 1 T x i = 1 A strategy profileσ is the collection of one strategy per agent, σ = (σ 1,...,σ N ) Example With Rock Paper Scissors games can be: x 1 = x 1,R = 0.2 x 1,P = 0.8 x 2 = x 2,R = 0.6 x 2,P = 0.0 x 1,S = 0.0 x 2,S = 0.4 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

12 Expected utility (1) Definition The expected utility of an agent i related to an action j is: U i x k k N,k i j where(a) j is the j th row of matrix A U i k N,k i x k is the vector of expected utilities of agent i The expected utility of an agent i related to a strategy x i is: x T i U i k N,k i x k Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

13 Expected utility (2) Example U 1 = x 1 = x 2 = The expected utilities related to each action of agent 1 are: = The expected utility related to the strategy of agent 1 is: [ ] = Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

14 Game equivalence Definition Given two games with utility functions U 1,...,U N and U 1,...,U N respectively, if, for every i N, there is an affine transformation between U i and U i such that U i = α iu i +β i A 1 where A 1 is a matrix of ones, then the two games are equivalent Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

15 Game equivalence Definition Given two games with utility functions U 1,...,U N and U 1,...,U N respectively, if, for every i N, there is an affine transformation between U i and U i such that U i = α iu i +β i A 1 where A 1 is a matrix of ones, then the two games are equivalent Example U 1 = U 1 = = Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

16 Solutions and solution concepts Definition Given: The strategy x i of each agent i The beliefˆx i j each agent i has over the strategy x j of agent j A solution is a pair(σ,µ), whereµis the set of agents beliefs, such that Rationality constraints: the strategies of each agent are optimal w.r.t. the beliefs Information constraints: the beliefs of each agent are somehow consistent w.r.t. the opponents strategies Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

17 Solutions and solution concepts Definition Given: The strategy x i of each agent i The beliefˆx i j each agent i has over the strategy x j of agent j A solution is a pair(σ,µ), whereµis the set of agents beliefs, such that Rationality constraints: the strategies of each agent are optimal w.r.t. the beliefs Information constraints: the beliefs of each agent are somehow consistent w.r.t. the opponents strategies Definition A solution concept defines the set of rationality and information constraints Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

18 Solution concept classification Non equilibrium solution concepts Dominance and iterated dominance Never best response and iterated never best response Maxmin strategy and minmax strategy Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

19 Solution concept classification Non equilibrium solution concepts Dominance and iterated dominance Never best response and iterated never best response Maxmin strategy and minmax strategy Equilibrium solution concepts without correlation Nash relaxations: conjectural equilibrium, self confirming equilibrium Nash Nash refinements: perfect equilibrium, proper equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

20 Solution concept classification Non equilibrium solution concepts Dominance and iterated dominance Never best response and iterated never best response Maxmin strategy and minmax strategy Equilibrium solution concepts without correlation Nash relaxations: conjectural equilibrium, self confirming equilibrium Nash Nash refinements: perfect equilibrium, proper equilibrium Equilibrium solution concepts with correlation One agent based correlation: leader follower/stackelberg/committment equilibrium Device based correlation: correlated equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

21 Dominance (1) Definition Action j A i is strictly dominated if there is a strategy x over A that, for every action of the opponents, provides an expected utility larger than action j e T j U i < x T U i where e j is a vector of zeros except for position j wherein there is 1 Example agent 1 Action C is dominated by action B agent 2 D E F A 4, 1 1, 2 1, 3 B 1, 4 4, 0 4, 1 C 0, 1 2, 5 2, 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

22 Dominance (2) Weakly dominance Action j A i is weakly dominated if there is a strategy x over A that, for every action of the opponents, provides an expected utility equal to or larger than action j Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

23 Dominance (2) Weakly dominance Action j A i is weakly dominated if there is a strategy x over A that, for every action of the opponents, provides an expected utility equal to or larger than action j Dominance and rationality No rational agent will play an action that is strictly dominated Strictly dominated actions can be safely removed from the game, never being played The application of strong dominance leads to a reduced game that is equivalent to the original one Weakly dominated actions could be played by agents Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

24 Dominance and mixed strategies Property Dominance with mixed strategies is stronger than with pure strategies Example agent 1 agent 2 D E F A 4, 1 1, 2 1, 3 B 1, 4 4, 0 4, 1 C 2, 1 2, 5 2, 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

25 Dominance and mixed strategies Property Dominance with mixed strategies is stronger than with pure strategies Example agent 1 Dominance in pure strategies agent 2 D E F A 4, 1 1, 2 1, 3 B 1, 4 4, 0 4, 1 C 2, 1 2, 5 2, 0 No action of the agent 1 is dominated by another action Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

26 Dominance and mixed strategies Property Dominance with mixed strategies is stronger than with pure strategies Example agent 1 Dominance in pure strategies agent 2 D E F A 4, 1 1, 2 1, 3 B 1, 4 4, 0 4, 1 C 2, 1 2, 5 2, 0 No action of the agent 1 is dominated by another action Dominance in mixed strategies Action C is dominated by x = [ ] Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

27 Dominance with more than two agents Example L R a 2, 2, 1 0, 3, 0 b 3, 0, 2 1, 1, 4 A L R a 2, 3, 0 0, 4, 1 b 3, 1, 2 1, 2, 0 B L R a 2, 1, 0 1, 0, 2 b 3, 3, 1 2, 3, 1 C Action a is dominated by action b Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

28 Dominance as a solution concept Comments Dominance does not require any assumption over the information available to each agent except for the knowledge of own utility Dominance prescribes what actions are to play and what are not to play independently of the opponents strategies Dominance does not prescribe any strategy over the non dominated actions We have an equilibrium in dominant strategies if dominance removes all the actions except one for every agent Example agent 2 S C agent 1 S 2, 2 0, 3 C 3, 0 1, 1 agent 1 agent 2 H T H 2, 0 0, 2 T 0, 2 2, 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

29 Iterated dominance Definition Under the assumption of complete information over the utility and common information over rationality and utilities, each agent can forecast the dominated actions of the opponents and iteratively remove her own actions Example agent 1 agent 2 D E F A 3, 2 2, 1 2, 0 B 0, 2 0, 5 3, 3 C 0, 1 1, 2 1, 4 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

30 Best response Definition The best response of agent i is an action that maximizes her expected utility given the strategies of the opponents as input BR i (σ i ) = arg max j A i et j U i x k where x k are given k N,k=i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

31 Best response Definition The best response of agent i is an action that maximizes her expected utility given the strategies of the opponents as input BR i (σ i ) = arg max j A i et j U i x k where x k are given Comments BR i (σ i ) can return multiple actions k N,k=i A rational agent will play only best response actions Any mixed strategy over best response actions is a best response Any non never best response action is said rationalizable Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

32 Never best response Definition A never best response of agent i is an action j such that there is not any opponents strategy profile such that action j is a best response j BR i (σ i ) σ i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

33 Never best response Definition A never best response of agent i is an action j such that there is not any opponents strategy profile such that action j is a best response j BR i (σ i ) σ i Comments No rational agent will play never best response actions Never best response actions can be safely removed Rationalizability requires each agent to know her own utilities, no assumption is required over the information on the opponents utilities and rationality Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

34 Never best response Definition A never best response of agent i is an action j such that there is not any opponents strategy profile such that action j is a best response j BR i (σ i ) σ i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

35 Never best response Definition A never best response of agent i is an action j such that there is not any opponents strategy profile such that action j is a best response j BR i (σ i ) σ i Comments No rational agent will play never best response actions Never best response actions can be safely removed When information on the utilities and rationality is complete and common, rationalizability can be iterated Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

36 Rationalizability and dominance (1) Comments Dominance and rationalizability are equivalent with two agents (the proof is by strong duality) With more than two agents, every dominated action is a never best response, but the reverse may not hold (rationalizability removes a larger number of actions than dominance) The main difference: Dominance is similar to rationalizability, but it implicitly assumes that the opponents correlate their strategy as a unique agent Rationalizability explicitly considers each opponent as a different uncorrelated agent If an action is dominated when the opponents can correlate is also dominated when they cannot If an action is dominated when the opponents cannot correlate, it may be not when they can Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

37 Rationalizability and dominance (2) Example L R a 0, 0, 0 0, 0, 0 b 8, 8, 8 0, 0, 0 L R a 0, 0, 0 8, 8, 8 b 0, 0, 0 0, 0, 0 L R a 4, 4, 4 0, 0, 0 b 0, 0, 0 4, 4, 4 L R a 3, 3, 3 3, 3, 3 b 3, 3, 3 3, 3, 3 A B C D Action D is not strictly dominated, but it is a never best response Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

38 Maxmin Assumptions An agent does not know anything about her opponents An agent aims at maximize her utility in the worst case (safety level) Definition A maxmin strategyσ of agent i is defined as: σ = arg max mine[u i ] σ i σ i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

39 Minmax Assumptions An agent knows the utility of the opponent An agent aims at minimize the opponent expected utility Definition A minmax strategyσ of agent i is defined as: σ = arg min maxe[u i ] σ i σ i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

40 Nash equilibrium (1) Assumptions Agents do not communicate before playing Agents know the utilities of the opponents and this information is common Definition A Nash equilibrium is a strategy profile(x 1,...,x n) such that: (x i )T U i j N,j i x j x T i U i j N,j i x j x i, i N Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

41 Nash equilibrium (1) Assumptions Agents do not communicate before playing Agents know the utilities of the opponents and this information is common Definition A Nash equilibrium is a strategy profile(x 1,...,x n) such that: Comments (x i )T U i j N,j i x j x T i U i j N,j i x j x i, i N In a Nash equilibrium, no agent can more by changing her strategy given that the opponents do not change (i..e, every x i is a randomization over best responses) Coalition deviations are not considered Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

42 Nash equilibrium (2) Definition A Nash equilibrium is a strategy profile(x 1,...,x n ) such that: (x i )T U i e T k U i k A i, i N Comments j N,j i x j j N,j i We can substitute x i (infinite constraints) with k A i ( A i constraints) because x T i U i j N,j i x j is a convex combination of different e T k U i j N,j i x j x T i U i j N,j i x j is smaller than or equal to max k e T k U i j N,j i x j since we cannot know what is k with the largest e T k U i j N,j i x j, we impose to be larger than equal to all the e T k U i j N,j i x j We obtain a finite number of constraints that is linear in the size of the game x j Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

43 Nash theorem Theorem Every finite game admits at least a Nash equilibrium in mixed strategies Comments The proof is by Brouwer fixed point theorem: a Nash equilibrium is a fixed point Pure strategies Nash equilibria may not exist (e.g., Matching penny) Multiple equilibria can coexist With continuous games, the things are more complicated (a continuous game may not admit any Nash equilibrium, neither in mixed strategies) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

44 Example (1) Pure strategy equilibrium agent 1 Pure strategy equilibrium agent 1 agent 2 D E F A 1, 3 2, 1 1, 0 B 3, 2 0, 5 2, 3 C 0, 1 1, 2 3, 3 agent 2 D E F A 6, 2 2, 1 1, 6 B 3, 2 3, 3 2, 3 C 0, 6 1, 2 3, 3 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

45 Example (2) Multiple pure strategy equilibria agent 1 No pure strategy equilibrium agent 1 agent 2 D E F A 6, 2 2, 1 1, 6 B 3, 2 3, 3 2, 3 C 0, 6 1, 2 9, 9 agent 2 D E F A 6, 2 2, 1 1, 6 B 3, 2 0, 3 2, 3 C 0, 6 1, 2 3, 3 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

46 Nash equilibrium and Pareto efficiency Example agent 2 S C agent 1 S 2, 2 0, 3 C 3, 0 1, 1 There is a unique Nash equilibrium(c, C) (C, C) is Pareto dominated by (S, S) (C, C) is the unique Pareto dominated strategy profile There is no relationship between Pareto dominance and Nash equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

47 Perturbed games (1) Perturbation Given a set of action A i, a perturbation over it corresponds to a probability function f i,j with j A i and j A i f i,j < 1 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

48 Perturbed games (1) Perturbation Given a set of action A i, a perturbation over it corresponds to a probability function f i,j with j A i and j A i f i,j < 1 Parametric perturbation Perturbation f i,j = f i,j (ǫ) withǫ [0, 1] Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

49 Perturbed games (1) Perturbation Given a set of action A i, a perturbation over it corresponds to a probability function f i,j with j A i and j A i f i,j < 1 Parametric perturbation Perturbation f i,j = f i,j (ǫ) withǫ [0, 1] Perturbed game Given a perturbation f i,j, a perturbed game is a game in which strategies are constrained as: i N, j A i : x i,j f i,j Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

50 Perturbed games (1) Perturbation Given a set of action A i, a perturbation over it corresponds to a probability function f i,j with j A i and j A i f i,j < 1 Parametric perturbation Perturbation f i,j = f i,j (ǫ) withǫ [0, 1] Perturbed game Given a perturbation f i,j, a perturbed game is a game in which strategies are constrained as: i N, j A i : x i,j f i,j Perturbation and Nash equilibrium The introduction of perturbation (i.e., a perturbation game) affects the set of Nash equilibria Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

51 Perturbed games (2) Example agent 1 agent 2 C D A 10, 10 0, 0 B 0, 0 1, 1 Perturbation: f 1,A = 0.2, f 1,B = 0.2, f 2,C = 0.2, f 2,D = 0.2 (A, C) and (B, D) are Nash equilibria without perturbation (0.8A+0.2B, 0.8C+ 0.2D) is a Nash equilibrium with perturbation: all the probability except for the perturbation is put on(a, C) (0.2A+0.8B, 0.2C+ 0.BD) is not a Nash equilibrium with perturbation: all the probability except for the perturbation cannot put on(b, D) Perturbation: f 1,A = 0.05, f 1,B = 0.05, f 2,C = 0.05, f 2,D = 0.05 (B, D) is a Nash equilibrium without perturbation (0.2A+0.8B, 0.2C+ 0.BD) is a Nash equilibrium with perturbation: all the probability except for the perturbation is put on(b, D) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

52 Perfect equilibrium (1) Definition A strategy profileσ is a perfect equilibrium if there is a f i,j (ǫ) such that, called σ (ǫ) a sequence of Nash equilibria for any ǫ [0,ǫ 0 ] of the associated perturbed games,σ (ǫ) σ as ǫ 0 Example agent 1 agent 2 C D A 1, 1 0, 0 B 0, 0 0, 0 For every f 1,A (ǫ) > 0, action D is not a best response For every f 2,C (ǫ) > 0, action B is not a best response (B, D) is a Nash equilibrium, but it is not perfect (A, C) is a perfect equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

53 Perfect equilibrium (2) Properties An equilibrium is perfect if it keeps to be a Nash equilibrium when minimally perturbed Every finite game admits at least a perfect equilibrium Every perfect equilibrium is a Nash equilibrium in which no weakly dominated action is played The vice versa (i.e., every Nash equilibrium in which no weakly dominated action is played is a perfect equilibrium) is true only for two player games There is not relationship between perfect equilibrium and Pareto efficiency We can safely consider only f i,j (ǫ) that are polynomial inǫ Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

54 Perfect equilibrium (3) Example C D A 1, 1, 1 1, 0, 1 B 1, 1, 1 0, 0, 1 E C D A 1, 1, 1 0, 0, 0 B 0, 1, 0 1, 0, 0 F F is weakly dominated for agent 3 D is weakly dominated for agent 2 (A, C, E) and (B, C, E) are Nash equilibria without weakly dominated actions (A, C, E) is not perfect Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

55 Perfect equilibrium (3) Example agent 1 agent 2 C D A 1, 1 10, 0 B 0, 10 10, 10 There are two pure strategy Nash equilibria (A, C) and(b, D) Actions B and D are weakly dominated The unique perfect Nash is (A, C) (B, D) Pareto dominates (A, C) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

56 Perfect equilibrium (5) Example agent 1 agent 2 A B agent 1 a 1, 1 0, 0 b 0, 0 0, 0 agent 2 A B C a 1, 1 0, 0-1,2 b 0, 0 0, 0 0,-2 c 2, 1 2, 0-2,-2 Without c and C, the unique perfect equilibrium is (a, A) With c and C, (b, B) is a perfect equilibrium The introduction of strictly dominated actions may change the set of perfect equilibria Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

57 Proper equilibrium (1) Perfection weakness The perfect equilibrium is sensible to weakly dominated actions Aim The design of a solution concept refining Nash equilibrium that is not sensible to weakly dominated actions Properness idea A proper equilibrium is a perfect equilibrium with a specific perturbation: given two actions j and k of agent i, if j provides a utility strictly larger than k, then perturbation f i,k is subject to f i,k ǫf i,j In other words The perturbation has the property that a good action must be played (due to perturbation) with probability larger than the probability of a bad action Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

58 Proper equilibrium (2) Properties Every game admits at least a proper equilibrium The proper equilibrium removes weakly dominated strategies with two player games With more agents, the proper equilibrium may not remove weakly dominated strategies Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

59 Correlated equilibrium (1) Assumptions Agents can correlate in some way Typically, a correlation device is considered that sends different signals to each agent Definition A correlated equilibrium is a tuple(v,π,σ), where v is a tuple of random variables v = (v 1,...,v n ) with respective domains D = (D 1,...,D n ),π is a joint distribution over v,σ = (σ 1,...,σ n ) is a vector of mappings σ i : D i A i, and for each agent i and every mapping σ i is the case that: d Dπ(d i, d i )U i (σ i (d i ),σ i (d i )) d D It is possible to limit strategies σ i to be pure π(d i, d i )U i (σ i (d i),σ i (d i )) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

60 Correlated equilibrium (2) Properties Every Nash equilibrium is a correlated equilibrium in which there is only one signal per agent A correlated equilibrium may be not a Nash equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

61 Leader follower equilibrium (1) Assumptions An agent, called leader, can announce (commit to) her strategy to the opponents The other agents, called followers, act knowing the commitment The announce must be credible Definition A leader follower equilibrium is a strategy profile in which the expected utility of the leader is maximized given that the followers act knowing the strategy of the leader Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120

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