Game Theory for Linguists

Transcription

1 Fritz Hamm, Roland Mühlenbernd 4. Mai 2016

2 Overview Overview 1. Exercises 2. Contribution to a Public Good 3. Dominated Actions

3 Exercises Exercise I Exercise Find the player s best response functions in the Prisoner s Dilemma and BoS. C D C 2,2 0,3 D 3,0 1,1 Prisoner s Dilemma B S B 2,1 0,0 S 0,0 1,2 Bach or Stravinsky B 1 (C) = {D}, B 1 (D) = {D} B 2 (C) = {D}, B 2 (D) = {D} B 1 (B) = {B}, B 1 (S) = {S} B 2 (B) = {B}, B 2 (S) = {S}

4 Exercises Exercise II Exercise Find the Nash equilibria of the following game. L C R T 2,2 1,3 0,1 M 3,1 0,0 0,0 B 1,0 0,0 0,0 B 1 (L) = {M}, B 1 (C) = {T }, B 1 (R) = {T, M, B} B 2 (T ) = {C}, B 2 (M) = {L}, B 1 (BR) = {L, C, R}

5 Exercises Exercise III Interlude: quadratic functions A function is quadratic if it takes the form f (x) = ax 2 + bx + c if a > 0 then its graph is U shaped if a < 0 its graph is an inverted U. Exercise Find the maximizer of the function x(α x) for any value of the constant α. x(α x) = x 2 + αx inverted U x(α x) = 0, iff x = 0 or x = α maximizer: x = α 2, f (x ) = α2 4

6 Exercises Review: Synergistic Relationship Two individuals are involved in a synergistic relationship. If both individuals devote more effort to the relationship, they are both better off. For any given effort of individual j, the return to individuals i s effort first increases, then decreases.specifically an effort level is a nonnegative number, and individual i s preferences (for i = 1, 2) are represented by the payoff function a i (c + a j a i ), where a i is i s effort level, a j is the other individuals effort level, and c > 0 is a constant.

7 Exercises Review: Synergistic Relationship Given: two players 1 and 2 strategies a i, a j N utility function u i (a i ) = a i (c + a j a i ) for all i {1, 2} Assume: c = ,0 0,1 0,0 0,-3 0,-8 1 1,0 2,2 3,2 4,0 5,-4 2 0,0 2,3 4,4 6,3 8,0 3-3,0 0,4 3,6 6,6 9, 4 4-8,0-4,5 0,8 4,9 8, Find the Nash equilibrium! (also for different c-values???)

8 Exercises Review: Synergistic Relationship The function a i (c + a j a i ) is an inverted U (since u i (a i ) = ai 2 + (c + a j )a i ) equals 0 when a i = 0 or when a i = c + a j Therefore the best response of each player i to a j is given by: b i (a j ) = 1 2 (c + a j)

9 Exercises Review: Synergistic Relationship To find the Nash equilibria solve the following system of equations: a 1 = 1 2 (c + a 2) a 2 = 1 2 (c + a 1) The unique (strict) Nash equilibrium is: (a 1, a 2 ) = (c, c).

10 Contribution to a Public Good Contribution to a Public Good w i wealth of person i c i (0 c i w i ) amount contributed by person i Players: two people Actions: Player i s set of actions is the set of her possible contributions (nonnegative numbers less than or equal to w i ), for i = 1, 2. Player i s preferences are represented by the payoff function u i (c 1, c 2 ) = v i (c 1 + c 2 ) c i or u i (c 1, c 2 ) = v i (c 1 + c 2 ) + w i c i

11 Contribution to a Public Good Contribution to a Public Good 1. Assume c 2 = 0 and assume that u 1 (c 1, 0) increases up to its maximum, then decreases. b 1 (0) denotes player 1 s best responds to c 2 = 0. This is the value of c 1 that maximimizes with 0 c 1 w 1. Assume 0 < b 1 (0) < w 1. u 1 (c 1, 0) = v 1 (c 1 ) c 1

12 Contribution to a Public Good Contribution to a Public Good 2. Now consider player 1 s best response to c 2 = k > 0. The best response is the value of c 1 that maximizes u 1 (c 1, k) = v 1 (c 1, k) c 1 We have Therefore u 1 (c 1 + k, 0) = v 1 (c 1 + k) c 1 k u 1 (c 1, k) = u 1 (c 1 + k, 0) + k

13 Contribution to a Public Good Contribution to a Public Good If k b 1 (0), then b 1 (k) = b 1 (0) k. This means that player 1 s best response decreases by k if player 2 s contribution increases from 0 to k. The same holds for player 2. If b 1 (0) > b 2 (0), the unique Nash equilibrium is (b 1 (0), 0). Player 2 contributes nothing. If b 1 (0) < b 2 (0) then the unique Nash equilibrium is (0, b 2 (0)): player 1 contributes nothing.

14 Contribution to a Public Good Contribution to a Public Good Only if b 1 (0) = b 2 (0) is there an equilibrium in which both people contribute. Any pair (c 1, c 2 ) such that c 1 + c 2 = b 1 (0) is a Nash equilibrium.

15 Dominated Actions Dominated actions You drive up to a traffic light. The left lane is free; in the right lane there is a car that may turn right when the light changes to green, in which case it will have to wait for a pedestrian to cross the side street. Assuming you wish to progress as quickly as possible, the action of pulling up in the left lane strictly dominates that of pulling up in the right lane.

16 Dominated Actions Dominated actions Definition (strict domination) In a strategic game with ordinal preferences, player i s action a i strictly dominates her action a i if u i (a i, a i ) > u i (a i, a i ) for every list a i of the other players actions where u i is a payoff function that represents player i s preferences. We say that the action a i is strictly dominated. A strictly dominated action is not used in any Nash equilibrium.

17 Dominated Actions Dominated actions L R T 1 0 M 2 1 B 1 3 L R T 1 0 M 2 1 B 3 2

18 Dominated Actions Dominated actions As you approach the red light in the above situation there is a car in each lane. The car in the right may, or may not, be turning right; if it is, it many be delayed by a pedestrian crossing the side street. The car in the left lane cannot turn right. In this case your pulling up in the left lane weakly dominates, though does not strictly dominate, your pulling up in the right lane. If the car in the right does not turn right, then both lanes are equally good; if it does, then the left lane is better.

19 Dominated Actions Definition (Weak domination) In a strategic game with ordinal preferences, player i s action a i weakly dominates her action a i if u i (a i, a i ) u i (a i, a i ) for every list a i of the other players actions and u i (a i, a i ) > u i (a i, a i ) for some list a i of the other players actions where u i is a payoff function that represents player i s preferences. We say that the action a i is weakly dominated. L R T 1 0 M 2 0 B 2 1

20 Dominated Actions Dominated actions Two candidates A and B, vie for office. Each of an odd number of citizens may vote for either candidate. (Abstention is impossible.) The candidate who obtains the most votes wins. (Because the number of citizens is odd, a tie is impossible.) A majority of citizens prefer A to win.

21 Dominated Actions Dominated actions Players: The citizens Actions: Each player s set of actions consists of voting for A and voting for B. Preferences: All players are indifferent among all action profiles in which a majority of players vote for A; all players are also indifferent among all action profiles in which a majority vote for B. Some players (a majority) prefer an action profile of the first type to one of the second type, and the others have the reverse preference.

22 Dominated Actions Dominated actions Claim: a citizen s voting for her less preferred candidate is weakly dominated by her voting for her favorite candidate.

23 Dominated Actions Symmetric games and symmetric equilibria I Definition A two-player strategic game with ordinal preferences is symmetric if the players sets of actions are the same and the player s preferences are represented by payoff functions u 1 and u 2 for which u 1 (a 1, a 2 ) = u 2 (a 2, a 1 ) for every action pair (a 1, a 2 ).

24 Dominated Actions Dominated actions A B A w,w x,y B y,x z,z Quiet Fink Quiet 2,2 0,3 Fink 3,0 1,1 Stag Hare Stag 2,2 0,1 Hare 1,0 1,1

25 Dominated Actions Dominated actions Definition An action profile a in a strategic game with ordinal preferences in which each player has the same set of actions is a symmetric Nash equilibrium if it is a Nash equilibrium and ai is the same for every player i. Left Right Left 1,1 0,0 Right 0,0 1,1

26 Dominated Actions Dominated actions X Y X 0,0 1,1 Y 1,1 0,0 Two Nash equilibria neither of which is symmetric.

27 Dominated Actions Conclusion of Sessions 1-3 a game describes a situation of multiple players that can make decisions (choose actions) that have preferences over interdependent outcomes (action profiles) preferences are defined by a utility function games can be defined by abstracting from concrete utility values, but describing it with a general utility function (c.f. synergistic relationship, contribution to public good) a solution concept describes a process that leads agents to a particular action a Nash equilibrium defines an action profile where no agent has a need to change the current action Nash equilibria can be determined by Best Response functions, or by iterated elimination of Dominated Actions