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## Basics on Game Theory

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**Basics on Game Theory**For Industrial Economics (According to Shy’s Plan)**The Reasons of Game Theory**• What • GT is the study of strategic interaction involving decisions among multiple actors • Why • Economic and political world is made of rules, actors and strategies • GT is the right frame for studying competition • What for • What are the alternatives of a game? • Can the behavior of other actors be predicted? • How to design the optimal strategies**What is a Game**• Normal Form Games • Players • Strategies • Payoffs • Extensive Form Games**Concepts**• Actions • Outcomes • Payoffs**Concepts**• In general: • Players i:1,...,n • Ai: Set of actions of player i. • ai: One action of player‘s set Ai • (a1, ... , an): Outcome • i(a1, ... , an): Payoff for player i, according to the actions of other players • Summarizing:**The Search for the Solution of a Game: Elimination of**Strictly Dominated Strategies: P2 Left Center Right Up P1 Down P2 Left Center Up P1 Down P2 Left Center Up P1**Dominant Actions (and one remark on notation)**• The payoff for player i, depends on his move and on the other’s moves (a1,a2,…, ai,…, an): • We represent it in the form: • An action a~ is dominant for one player i, if:**Nash Equilibrium**• If the solution of the game is unique, it is a Nash Equilibrium. • Example • Definition of NE: P2 Left Center Right Up P1 Middle Down**Games with Multiple Equilibria**• The Battle of the Sexes I He Opera Football Opera She Football**Games without NE**• The Battle of the Sexes II He Opera Football Opera She Football**The Best Response Function**He Opera Football Opera She Football**Pareto Efficient Outcomes**• The Prisoners’ Dilemma P2 Defect Cooperate -1,-1 -9,0 Defect P1 0,-9 -6,-6 Cooperate**Some Questions**• What is a normal form game? • What is a strictly dominated strategy? • What is a NE in a normal form game? • What are the advantages and the shortcomings of GT in the prediction of the strategic behavior?**Exercise**• In the next game in normal form, which strategies survive to the elimination of strictly dominated strategies? P2 Left Center Right Up P1 Middle Down**Payoffs P(1):**1 2 0 3 0 Payoffs P(2): 1 2 1 Extensive Form Games 1 L R 2 2 L´ R´ L´ R´ Characteristics: 1) Moves occur in sequence 2) All the previous moves are observed before choose the next one 3) Payoffs are common knowledge among the players**Backward Induction and NE**1 L R 2 (2,0) L´ R´ 1 (1,1) L´´ R´´ (3,0) (0,2) Induction: 1. Step 3. P1 chooses L´´ with u1 = 3 instead of R´´ with u2 = 0 2. Step 2. P2 anticipates that if the game reaches level 3, then P1 chooses L´´ therefore u2 = 0. P2 chooses L´ with u2 = 1. 3. Step 1. P1 anticipates that if the game reaches level 2, then P2 chooses L´ and therefore u1 = 1. Then, P1 chooses L with u1 = 2.**1**L R 2 2 L´ R´ L´ R´ Payoffs P(1): 1 2 0 3 0 Payoffs P(2): 1 2 1 Strategies in Extensive Form Games One strategy is a complete plan of actions specifying a feasible action for each move in each contingency for which he can be called upon to act. P1 has 2 actions A{L,R} SP1 coincides with A{L,R} P2 has 2 actions A{L,R} but 4 strategies. Strategy 1: If P1 plays L, then play L´, if P1 plays R, then play L´: (L´,L´). Strategy 2: If P1 plays L, then play L´, if P1 plays R, then play R´: (L´,R´). Strategy 3: If P1 plays L, then play R´, if P1 plays R, then play L´: (R´,L´). Strategy 4: If P1 plays L, then play R´, if P1 plays R, then play R´: (R´,R´).**1**L R 2 2 L´ R´ L´ R´ Payoffs P(1): 1 2 0 3 0 Payoffs P(2): 1 2 1 NE of Extensive Form Games Which strategy is the NE of the game? Strategy 1: (L´,L´) Strategy 2: (L´,R´) Strategy 3:(R´,L´) Strategy 4: (R´,R´)**1**L R 2 2 L´ R´ L´ R´ Payoffs P(1): 1 2 0 3 0 P2 Payoffs P(2): 1 2 1 (L´,L´) (L´,R´) (R´,L´) (R´,R´) (L) P1 (R) Normal Form from Extensive Form**Subgame Perfect Nash E.**Definition: A NE is Subgame Perfect if the strategies of the players constitute a NE in each subgame. Algorithm for Identifying a SPNE: Identify all the smaller subgames having terminal nodes in the original tree. Replace each subgame for the payoffs of one of the NE. The initial nodes of the subgame are now the terminal nodes of the new truncated tree.**1**L R 2 2 L´ R´ L´ R´ Between 1 and 2, P1 prefers to play R. Payoffs P(1): 1 2 0 3 0 Payoffs P(2): 1 2 1 Subgame 1 Subgame 2 Subgame Perfect Nash E. Example: SPNE = ( , ) R` L`**1**L R SPNE = (R`,L`) 2 2 L´ R´ L´ R´ Backward Induction = (R,L`) Payoffs P(1): 1 2 0 3 0 Payoffs P(2): 1 2 1 NE and Subgame Perfect NE Subgame Perfect Nash Equilibrium vs. Simple NE SPNE is more powerful than NE, for solving Imperfect Information Games: