Basics of Game Theory

Transcription

1 Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering isa University April - May, 2010 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

2 Games Taxonomy A game can be: ooperative Non-cooperative Strategic Extensive erfect information Imperfect information omplete information Incomplete information and many more types... G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

3 Outline Bayesian Games Motivation examples Nash equilibrium General definition G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

4 onsider the Battle of Sexes game. Tim 3, 1 0, 0 0, 0 1, 3 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

5 onsider the Battle of Sexes game. Tim 3, 1 0, 0 0, 0 1, 3 With complete information each player knows perfectly the game is playing: who the other players are; what their possible strategies are; and what payoff will result for each player for any combination of moves. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

6 Definition In many situations a player may not be fully informed about his opponents; a player may not know how well opponents are informed. These situations can be modeled as strategic games with incomplete information. A strategic game with incomplete information is called Bayesian game. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

7 onsider a variant of the Battle of Sexes game. Tim does not know whether prefers to go out with him or to avoid him. Tim 3, 1 0, 0 Tim 3, 0 0, 1 0, 0 1, 3 0, 1 1, 0 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

8 onsider a variant of the Battle of Sexes game. Tim does not know whether prefers to go out with him or to avoid him. Tim 3, 1 0, 0 Tim 3, 0 0, 1 0, 0 1, 3 0, 1 1, 0 On the contrary, knows Tim preferences. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

9 Suppose Tim believes that wishes to meet him with probability 2/3. How can we model this situation in strategic form? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

10 Suppose Tim believes that wishes to meet him with probability 2/3. How can we model this situation in strategic form? 2/3 1/3 1 1 Tim 3,1 0,0 3,0 0,1 0,0 1,3 0,1 1,0 State yy State yn G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

11 Suppose Tim believes that wishes to meet him with probability 2/3. How can we model this situation in strategic form? 2/3 1/3 1 1 Tim 3,1 0,0 3,0 0,1 0,0 1,3 0,1 1,0 State yy State yn knows the game is playing. Tim believes that with probability 2/3 plays the game on the right. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

12 Suppose Tim believes that wishes to meet him with probability 2/3. How can we model this situation in strategic form? 2/3 1/3 1 1 Tim 3,1 0,0 3,0 0,1 0,0 1,3 0,1 1,0 State yy State yn knows the game is playing. Tim believes that with probability 2/3 plays the game on the right. It is like a three player game. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

13 To choose an action rationally, Tim has to form a belief about his actions. Since probabilities are involved expected payoffs must be computed. This happens also if we are only interested in pure equilibria. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

14 To choose an action rationally, Tim has to form a belief about his actions. Since probabilities are involved expected payoffs must be computed. This happens also if we are only interested in pure equilibria. Assume Tim believes that the type who wishes to meet him will choose. Assume Tim believes that the type who wishes to avoid him will choose. What is his expected payoffs for and? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

15 What is Tim s expected payoff for? What is Tim s expected payoff for? u 1(,(, )) = = 2 u 1(,(, )) = = 1 3 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

16 Similar computations leads to the following result. (,) (,) (,) (,) Tim /3 2/3 1 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

17 Nash equilibrium A Nash equilibrium models a situation in which each player s beliefs about the other player s actions are correct, and each player acts optimally, given her beliefs. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

18 Nash equilibrium A Nash equilibrium models a situation in which each player s beliefs about the other player s actions are correct, and each player acts optimally, given her beliefs. A Nash equilibrium is found looking for the best response. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

19 Nash equilibrium How apply the best response procedure to the above Bayesian game? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

20 Nash equilibrium How apply the best response procedure to the above Bayesian game? Treat the game like a three player game in which Tim (player 1) depends on the two s (players 2 and 3); each depends only on Tim. Find a triple of actions, one for Tim and one for each, such that the action of Tim is optimal given those of ; the action of each type of is optimal given that of Tim. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

21 Nash equilibrium Assume Tim chooses while (, ). Is (,(, )) a Nash equilibrium or not? Tim y (,) (,) (,) (,) ,1 0,0 3,0 0,3 0 1/3 2/3 1 0,0 1,3 0,1 1,0 State yy State yn G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

22 Nash equilibrium Assume Tim chooses while (, ). Is (,(, )) a Nash equilibrium or not? Tim y (,) (,) (,) (,) ,1 0,0 3,0 0,3 0 1/3 2/3 1 0,0 1,3 0,1 1,0 State yy State yn Given (, ), is the best response for Tim. Given, is the best response for 1 and is the best response for 2. Then, (,(,)) is a Nash equilibrium. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

23 Nash equilibrium Assume Tim chooses while (, ). Is (,(, )) a Nash equilibrium or not? Tim y (,) (,) (,) (,) ,1 0,0 3,0 0,3 0 1/3 2/3 1 0,0 1,3 0,1 1,0 State yy State yn Given (, ), is the best response for Tim. Given, is the best response for 1 and is the best response for 2. Then, (,(,)) is a Nash equilibrium. heck that (,(, )) cannot be a Nash equilibrium. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

24 Nash equilibrium Does really have to plan in both cases? NO! G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

25 Nash equilibrium Does really have to plan in both cases? NO! The reason is that Tim has to form a belief. Equilibrium is achieved only if this belief is correct. Then, equilibrium actions must be interpreted as a proof of this correctness. They don t have to be intended as a plan of actions of. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

26 onsider now. 2/3 1/ ,1 0, ,0 0,3 0,0 1,3 0,1 1,0 Tim 1 2 State yy 2/3 1 2 State yn 1/3 0,1 3,0 0,0 3,3 1,0 0,3 1,1 0,0 State ny State nn G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

27 heck that ((, ),(, )) and ((, ),(, )) are Nash equilibria. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16

28 heck that ((, ),(, )) and ((, ),(, )) are Nash equilibria. Tim y Tim n (,) (,) (,) (,) (,) (,) (,) (,) /3 2/ /3 1/3 0 y n (,) (,) (,) (,) (,) (,) (,) (,) 1 1/2 1/ /2 1/ /2 3/ /2 3/2 0 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 16