Expectations & Randomization Normal Form Games Dominance Iterated Dominance. Normal Form Games & Dominance

Transcription

1 Normal Form Games & Dominance

2 Let s play the quarters game again We each have a quarter. Let s put them down on the desk at the same time. If they show the same side (HH or TT), you take my quarter. If they show opposite sides (HT or TH), I take yours. Q: What is your expected payoff for playing H? A: It depends on what you believe I will choose. Suppose that you think I will choose H with probability p. If you play H, you expect to win (50) with probability p and lose with probability 1 p So your expected value from H is EV [H] = p 50 + (1 p) 0 = 50p Modeling choice under uncertainty: beliefs form, expected payoff calculated, outcomes weighted by perceived likelihood

3 Let s play the quarters game again We each have a quarter. Let s put them down on the desk at the same time. If they show the same side (HH or TT), you take my quarter. If they show opposite sides (HT or TH), I take yours. But this time you have a third option (besides H and T): let your neighbor flip the coin for you. What do you choose? (H,T, or Flip) Notice: Uncertainty can come from randomizing own strategy (mixed strategy), not just strategic uncertainty (or external randomness)

4 Part I: Games in Normal Form Definition A strategy is a complete contingent plan for a player in the game. Example: Pick the highest integer. What are the strategies of each player? General notation: individual strategy space = S i, individual strategy = s i S i. A strategy profile: s S, where S = S 1 S 2 S n Notation: i and i. s = (s i, s i ). Another element: Payoff function u i : S R Definition A game in normal form (also called strategic form) consists of a set of players, along with strategy spaces and payoff functions for each player.

5 Matrix Games Two-player, finite (as in S is finite) normal form games can be described succinctly with bi-matrices. Take a look at your handout. Let s verify that the first matrix game satisfies the requirements of the normal form. Ok, before starting to analyze this game, let s try playing one of these...

6 Let s Play Let s play (g) for real money (see handout) For (g): odd perm number choose A, B, or C; even perm number choose X, Y, or Z. Two separate clicker votes: first row players, then column players. Now: I will select participants by drawing two cards for each game, pay in $ according to strategies chosen Next time: I will select two more people and pay according to clicker vote from today. What kind of reasoning did you use?

7 How do clickers benefit us? Make it easy to play (some) in-class games Can help me see what you ve learned Can help you see whether or not you understand Can help you see how far you ve come Allow you to give me feedback/opinions Also: attendance/participation Make sure your clicker is registered, bring it every class.

8 Back to game a) How do you win at this game? Ask Dilbert What s going on in this game?

9 Three concepts at play Concepts Beliefs: What Player i thinks all the other players will do (θ i S i ). Best response: A strategy is a best response if it is at least as good as any other strategy, given beliefs. A strategy is dominated if there is another strategy that is better for every set of beliefs. In other words, if Y is better than X no matter what you think other players will choose, then X is dominated by Y. No rational person would ever choose a strategy that is dominated

10 Solution Concept: Dominance Assumes rationality: no one would ever choose an action this is dominated by another. Apply to PD by crossing out all rows or columns that are dominated Now let s try to solve b) Does each player in game (b) have a dominant strategy? NO! b) is not dominance solvable

11 What s the problem? Need to further narrow down choices Rationality is too weak an a assumption Stronger: assume rationality is common knowledge Rational, know you are, know you know, know you know you know, etc. Eliminate strategies through iterative process Our second solution concept: Iterated Elimination of Dominated Strategies

12 Iterated Dominance Also known as Rationalizability Strategies that survived iterated elmination are called rationalizable Try on b) Now we get a unique outcome. Yay!

13 Iterated Dominance: Weaknesses? Is iterated dominance THE ULTIMATE SOLUTION CONCEPT? To see, let s return to game c). Is c) dominance solvable? NO! We can get more predictive power with stronger assumptions. Do we want this?

14 Iterated Dominance: Are we missing anything? Let s try d) Seems like no dominance going on. Are we missing anything? Free your mind and dominance will follow If we enrich the strategy space to allow mixed strategies, then B is dominated By what? (1/2, 0, 1/2) Now we continue iterated elimination A strategy may be dominated by a combination of other strategies

15 Player 1 s expected payoff, as a function of beliefs about Player 2 s strategy

16 Mixed strategies and best responses Let s look at e) now Is B dominated? No, not even by mixed strategies Is it ever a best response? Not to X or Y But it is a BR to (1/2, 1/2) (one example) If a strategy is a best response to some beliefs, it can t be dominated A strategy may be a best response (therefore undominated) even if it is not the best response to any pure strategy