Introduction to Game Theory I

Transcription

1 Nicola Dimitri University of Siena (Italy) Rome March-April 2014

2 Introduction to Game Theory 1/3 Game Theory (GT) is a tool-box useful to understand how rational people choose in situations of Strategic Interaction, which often occur in Economics. GT has also been applied and investigated in Biology, Linguistics, Philosophy, Logic, Applied Neuroscience, Political Science, Psychology. Strategic interaction. Individuals, players, in a small group, are aware that the final outcome of an action that they take depends on everyone s action. Strategic reasoning is everyday thinking

3 Introduction to Game Theory 2/3 Importance of Expectations: since players welfare depends on everyone s action, before choosing they form expectations on the others choice. Examples: product pricing and quantity setting by firms, repeated relationships (procurement), traffic, language, etc. Games. Games can be divided into two classes: noncooperative and cooperative games

4 Introduction to Game Theory 3/3 Non-Cooperative Games: the class of games modeling situations where players agreements can not be enforced by a third party (no formally written contracts) Cooperative Games: the class of games modeling situations where players agreements can be enforced by a third party (written contracts) We shall only concentrate on Non-Cooperative Games

5 Non-Cooperative Games 1/2 Non-Cooperative Games (NCG): NCG study situations where individuals are rational, in the sense that they maximize their own welfare (benefits), and may try to coordinate their own actions stipulating (informal) agreements that, however, can not be enforced by a third party (court). Strategic (Static) and Extensive Form (Dynamic) Games : NCG in turn can be divided in strategic, and extensive, form games

6 Non-Cooperative Games 2/2 Strategic (Static) Games (SG) A class of games where players choose independently of each other. Namely, when choosing their actions they have no information on the rivals choice. They may only have expectations. Extensive Form (Dynamic) Games (EFG): A class of games where some players choose having information on the actions chosen by rivals.

7 Strategic Games (SG) 1/13 We first describe more precisely a SG and then would make predictions on which actions players could plausibly choose Strategic Game is described by: (i) the number (set) of players N. (ii) the actions, or strategies, each player can take in that situation. (iii) each player s payoff Note Since the situation is one of strategic interaction a player s payoff depends upon the actions taken by all players, and not just by her own action.

8 Strategic Games (SG) 2/13 Classic Example: Cournot Duopoly Two large firms, A and B, are the only ones producing a certain good in the market. Based on their knowledge of market demand for that good, while setting the same product price to maximize their profits they have to decide how much to produce. If they truly compete, namely they do not collude, both A and B might have to decide their production level without knowing how much the other firm produces. Therefore, in order to choose profit maximizing quantities each firm can at most form expectations on the production level of the other firm. The model proposed by Augustine Cournot ( ) Cournot Duopoly could then be modeled as a SG, where the two firms are the players, actions are given by their possible production levels and their payoffs by the firms profits, which depend upon both firms production level.

9 Strategic Games (SG) 3/13 Game Solutions Upon having described a SG we want to make predictions on what actions players could plausibly choose in the game. In NCG there are two main solution concepts: (i) Action Dominance (ii) Nash Equilibrium

10 Strategic Games (SG) 4/13 Action Dominance For a player A an action, say a*, that she can take is said to dominate all her other possible actions if, independently of the opponents choice, a* provides A with a payoff (benefit) higher that any other action a she could have taken.

11 Strategic Games (SG) 5/13 Prisoner s Dilemma (PD) Cooperate Defect Cooperate 2,2 0,3 Defect 3,0 1,1

12 Strategic Games (SG) 6/13 Prisoner s Dilemma (PD) In the Prisoner s Dilemma choosing to defect is a (strictly) dominating action (over cooperating ) for both players. If players are self-interested there is no reason for them to cooperate, and both players defecting is our prediction in this game However, if they both cooperate they would both be better-off than when they defect. Excess of rationality is not efficient.

13 Strategic Games (SG) 7/13 Nash Equilibrium (NE) However, most games dot not have dominant actions. In this case predictions on players choice are made by making use of the more sophisticated notion of Nash Equilibrium. Nash Equilibrium Suppose there are two players, A and B, choosing (respectively) actions a* and b*. Then the pair of actions (a*,b*) are a NE if a* is best for A against b* and b* is best for B against a*. Namely, if (a*,b*) is a NE then A has no incentive to use a strategy different from a* if she thinks B will choose b*, and viceversa.

14 Strategic Games (SG) 8/13 Bach or Strauss (BoS) B S B 2,1 0,0 S 0,0 1,2

15 Strategic Games (SG) 9/13 Bach or Strauss (BoS); multiple NE In the game BoS (also called Battle of Sexes ) players have no dominant strategies. However, the game has two Nash Equilibria: (B,B) and (S,S); the problem with multiple equilibria is uncertain prediction. BoS is a coordination game, where players are better-off when choosing the same action. Having said so, the notion of NE per se does not help selecting between the two possible predictions.

16 Strategic Games (SG) 10/13 Question: when (strictly) dominating actions exist for all players, do they provide the same predictions as NE? Answer: YES. In particular, if in a game each player has a strictly dominating action then those actions represent the unique NE of the game (like in the Prisoner s Dilemma)

17 Strategic Games (SG) 11/13 Question: is strict dominance the only notion of dominance? Answer: NO. There could be games with, so called, weakly dominant strategies.

18 Strategic Games (SG) 12/13 A game with weakly dominant actions and two NE. In the game R and D are weakly dominant for the players, and (D,R) is a NE. However, unlike strict dominance, weak dominance does not guarantee a unique NE. Indeed, (U,L) is also a NE, though inefficient (QWERTY keyboard). L R U 0,0 0,0 D 0, 0 1,1

19 Strategic Games (SG) 12/13 Not all games have a Nash Equilibrium, which means that we can not always use NE to make predictions. For example. In this cases, we could the generalized notion of Mixed Strategy NE. L R U 1,-1-1,1 D -1, 1 1,-1