Game Theory DR. ÖZGÜR GÜRERK UNIVERSITY OF ERFURT WINTER TERM 2012/13. Strict and nonstrict equilibria

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1 Game Theory 2. Strategic Games contd. DR. ÖZGÜR GÜRERK UNIVERSITY OF ERFURT WINTER TERM 2012/13 Strict and nonstrict equilibria In the examples we have seen so far: A unilaterally deviation from Nash equilibrium leads to a worse outcome for the deviant player 2 Nash equilibrium, however, requires that a deviation from the equilibrium action should not lead to a no better outcome (i.e., not necessarily to a worse outcome) In some games, a player may be indifferent between her equilibrium action and some other action

2 Example: Nonstrict equilibrium This game has a unique Nash equilibrium: (T, L) 3 If P2 chooses her eq.-action L, P1 could equally choose T or B P2 L M R If P1 deviates to from T to B, she is no worse than in equilibrium P1 T 1, 1 1, 0 0, 1 B 1, 0 0, 1 1, 0 Hence, the Nash eq. (T, L) is a nonstrict equilibrium Strict Nash equilibrium An equilibrium is strict if each player s equilibrium action is better than all her other actions, given other players actions More formally: An action profile a* is a strict Nash equilibrium if for every player i we have u i (a*)>u i (a i,a -i *) for every action a i a i * of player i u i : Payoff function representing player i s preferences a i *: Player i s equilibrium action a -i *: Other players equilibrium i actions In plain words: The payoff from the equilibrium action is higher than the payoffs from all other possible actions given that other players stick to their equilibrium actions 4

3 What is a best response? Which actions of a player i yield the highest payoffs given the actions of other players? 5 In BoS example: Bach is the best action of P1 if P2 chooses Bach; Stravinsky is the best action if P2 chooses Stravinsky P1 In this game: both B and T are best actions of P1 if P2 chooses L P2 Bach Stravinsky Bach 2, 1 0, 0 Stravinsky 0, 0 1, 2 P2 L M R P1 T 1, 1 1, 0 0,1 B 1, 0 0, 1 1, 0 We define the function B i by Best response functions B i (a -i )={a i in A i : u i (a i,a -i ) u i (a i,a -i ) for all a i in A i } : i i i i i i i i i i i i 6 any action in B i (a -i ) is at least as good as for player i as every other action of player i when the other players actions are given by a -i a i a i a -i : Player i s (best response) action : Player i s other actions : Other players actions

4 Best response functions BR function is set-valued: It associates a set of actions with any list of other players actions Every member of the set B i i( (a -i ) is a BR of player i to a -i 7 In BoS game: The set B i (a -i ) consists of a single action for every list a -i of actions of the other players In other games the set B i (a -i ) may contain more than one action for some lists a -i of other players actions -i p y Using BR functions to define Nash eq. The action profile a* is a Nash equilibrium of a strategic game (with ordinal preferences) if and only if every player s action is a best response to the other players actions: a i * in B i (a -i *) for every player i 8 If each player i has a single best response to each list a -i of the other players actions we can write a collection of n equations a i * = b i (a -i *) for every player i n: the number of players with b i (a -i ) we denote the single element of the set B i (a -i )

5 Using BR functions to define Nash eq. In a game with two players, 1 and 2; these equations are: a 1 * = b 1 (a 2 *) a 2 * = b 2 (a 1 *) 9 In a two-player game in which each player has a single best response (BR) to every action of other player, (a 1 *, a 2 *) is a Nash equilibrium if and only if a 1 * is a BR to a 2 * and a 2 * is a BR to a 1 * Using BR to find Nash eq. Find the best response functions 10 Find the action profiles that satisfy a i * in B i (a -i *) for every player i Example: Find BR of P1 to each action of P2 Find BR of P2 to each action of P1 Nash equilibria: (M,L) and (B,R) P2 L C R T 1, 2* 2, * 1 1, * 0 P1 M 2, * 1 * 0, 1 * 0, 0 B 0, 1 0, 0 1, * 2*

6 Example: A synergistic relationship Two individuals are involved in a relationship (players) If both individuals put more effort to the relationship, they are both better off For any given effort of individual id j, the return to individual id i s effort first increases, then decreases The individual i s preferences (for i = 1, 2) are represented by the payoff function a i (c + a j a i ) a i 0 is i's effort level l (set t of actions) ) a j 0 is the other individual's effort level c > 0 is a constant 11 Finding the BR functions Suppose a j is given, what is the highest payoff of player i? 12 To find the maximum, we have to set the derivative of player i s payoff function a i (c + a j a i ) with respect to a i equal to zero The derivative is: c + a j 2a i Setting equal to zero and rearranging: a i = ½ (c + a j ) For given a j, player i s payoff function is maximized at a i = ½ (c + a j) The BR of player i is: b i (a j ) = ½ (c + a j )

7 P1 s actions plotted on the horizontal axis Plotting the BR functions 13 P1 s BR function associates an action for P1 with every action of P2 Interpretation t ti of b1: Take a point on the vertical axis,,g go to b1, read down the horizontal axis to find b 1 (a 2 ) Finding the Nash equilibria At a point (a 1, a 2 ) where the BR functions intersect in the figure, we have a 1 = b 1 (a 2 ), because (a 1, a 2 ) is on the graph of b1, and a 2 = b 2 (a 1 ), because (a 1, a 2 ) is on the graph of b2 14 Thus any such point (a 1, a 2 ) is a Nash equilibrium In this game the BR functions intersect at a single point, so there is one Nash equilibrium In general, they may intersect more than once; every point at which they intersect is a Nash equilibrium

8 Finding the Nash equilibria To find the point of intersection of the BR functions precisely, we can solve the two equations: a 1 = 1/2 (c + a 2 ) a 2 = 1/2 (c + a 1 ) Substituting the second equation in the first, we get a 1 = ½ (c + 1/2 (c + a 1 )) = 3/4 c +1/4 a 1, so that a 1 = c Substituting this value of a1 into the second equation, we get a 2 = c We conclude that the game has a unique Nash equilibrium (a 1, a 2 ) = (c, c) 15 Dominated actions: Strict domination In any game, a player s action strictly dominates another action if it is superior, no matter what the other players do 16 In the PD game, fink strictly dominates quiet Suspect 1 Suspect 2 quiet fink quiet 2, 2 0, 3 fink 3, 0 1, 1 In BoS, neither action strictly dominates the other P1 P2 Bach Stravinsky Bach 2, 1 0, 0 Stravinsky 0, 0 1, 2

9 Dominated actions 17 A strictly dominated action is never a best response to some other action such an action is not used by any Nash eq. Example: The action quiet in the PD game An action a that strictly dominates a, may itself be strictly dominated by another action L R T 1, 0, Example: M strictly dominates T but M is strictly dominated by B M 2, 1, B 3, 2, Dominated actions: Weak domination An action weakly dominates another action if it is at least as good as the other action, no matter what the other players do, and is better than the second action for some actions of the other players 18 Example: M weakly dominates T B weakly dominates M B strictly dominates T L R T 1, 0, M 2, 0, B 2, 1,

10 Dominated actions & (non)strict equilibria In a strict Nash equilibrium no player s equilibrium action is strictly dominated by nonequilibrium actions 19 In a nonstrict Nash equilibrium, however, an equilibrium action may be weakly dominated Examples: In both games (B,B) and (C,C) are the Nash eq. In both games B weakly dominates C B C B C B 1, 1 0, 0 B 1, 1 2, 0 C 0, 0 0, 0 C 0, 2 2, 2