1 / 18

Game Theory and Strategy

Game Theory and Strategy. Week 2 – Instructor: Dr Shino Takayama. Agenda for Week 2. Strict and Nonstrict Equilibria Best Response Functions Dominated Actions Strict Domination Weak Domination Illustration Contributing to a public good Voting Symmetric Games. Nash Equilibrium.

gloverr
Download Presentation

Game Theory and Strategy

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Game Theory and Strategy Week 2 – Instructor: Dr Shino Takayama

  2. Agenda for Week 2 • Strict and Nonstrict Equilibria • Best Response Functions • Dominated Actions • Strict Domination • Weak Domination • Illustration • Contributing to a public good • Voting • Symmetric Games

  3. Nash Equilibrium • The action profile a* in a strategic game with ordinal preferences is a Nash equilibrium if for every player i, ui(a*) ≥ ui(ai, a-i*), for every action aiof player i, where ui is a payoff function that represents player i’s preferences.

  4. Strict and nonstrict equilibria • The definition of Nash equilibria only requires that the outcome of a deviation be no better for the deviant than the equilibrium outcome. • An equilibrium is strict if each player’s equilibrium action is better than all her other actions, given the other players’ actions. • The action profile a* is a strict Nash equilibrium if for every player i, ui(a*) > ui(ai, a-i*), for every action ai ≠ai*of player i.

  5. Example • A game with a unique Nash equilibrium, which is not a strict equilibrium: L M R T (1, 1) (1, 0) (0, 1) B (1, 0) (0, 1) (1, 0)

  6. Best Response Functions • We define the function Bi by Bi(a-i) = {ai in Ai : ui(ai, a-i) ≥ ui(ai’, a-i) for all ai’ in Ai }. • We call Bithe best response function of player i. • The action profile is a Nash equilibrium of a strategic game if and only if ai* is in Bi(a-i*) for every player i.

  7. Find a Nash Equilibrium: L C R T (1, 2) (2, 1) (1, 0) M (2, 1) (0, 1) (0, 0) B (0, 1) (0, 0) (1, 2) • L M; C  T; R  T, B • T L; M  L, C; B  R

  8. Dominated Actions: Strict Domination • Player i’s action ai’’ strictly dominates ai’ if ui(ai’’, a-i) > ui(ai’, a-i), for every list a-iof the other players’ actions. • We say that the action ai’ is strictly dominated.

  9. L R T 1 0 M 2 1 B 1 3 L R T 1 0 M 2 1 B 3 2 Example for Strict Domination * Only player 1’s payoffs are given.

  10. Dominated Actions: Weak Domination • Player i’s action ai’’ weakly dominates ai’ if ui(ai’’, a-i) ≥ ui(ai’, a-i), for every list a-iof the other players’ actions with strict inequality for some list. • We say that the action ai’ is weakly dominated.

  11. Example for Weak Domination L R T 1 0 M 2 0 B 2 1 * Only player 1’s payoffs are given.

  12. Nash Equilibrium and Strict & Nonstrict Domination • A strictly dominated action is not a best response to any actions of the other players. • When looking for the Nash equilibrium, we can eliminate from all strictly dominated actions. • A weakly dominated action can be in a nonstrict Nash equilibrium.

  13. Illustration 1: contributing to a public good • Denote person i’s wealth by wi, and the amount she contributes to the public good by ci (0 ≤ ci ≤ wi). She spends her remaining wealth wi - ci on private goods. The amount of the public good is equal o the sum of the contributions. • Suppose that person preferences are represented by the payoff function vi(c1 + c2) + wi - ci, where vi is an increasing function.

  14. Illustration 1: Analysis • The strategic game: • Players: The Two People • Actions: Nonnegative numbers less than or equal to wi • Preferences are represented by the payoff function: ui(c1, c2) = vi(c1 + c2) – ci . Suppose that u1(c1, 0) increases up to its maximum and then decreases. • Consider the two cases: • c2 = 0; • c2 = k.

  15. Graphical Illustration 1 u1(c1, k) k k u1(c1, 0) 0 b1(k) b1(0) c1

  16. Graphical Illustration 2 c2 Nash equilibrium is (b1(0), 0): player 2 contributes nothing. b1(c2) b2(0) b1(c2) b2(c1) 0 b2(c1) b1(0) c1

  17. Illustration 2: voting • Players: The two suspects • Actions: {Voting for A, Voting for B} • Preferences: All players are indifferent among all action profiles in which : • a majority of players vote for A; • a majority of players 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.

  18. Symmetric Games • A two-player strategic game with ordinal preferences is symmetric if the players’ sets of actions are the same and the players’ preferences are represented by payoff functions u1 and u2 for which u1(a1, a2) = u2(a2, a1), for every action pair (a1, a2) . • An action profile ai* in a strategic game 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.

More Related