Chapter 13 Game Theory and Competitive Theory What is game theory?
ECON191 (Spring 2011) 18 & 19.5.2011 (Tutorial 12) Chapter 13 Game Theory and Competitive Theory What is game theory? Game theory is a method for modeling decision making when decision interact A game is characterized by (i) The set of players (ii) The strategy set (the set of feasible actions) - A strategy is a complete plan of action, that tells the player what to do every time where he has the move. (iii)The payoffs of the players - Payoff of a player depends not only on his own strategy, but also the strategy of the other player (interdependence). In game theory, we assume players are rational and they are only interested in their own payoffs. Dominant strategy and Nash Equilibrium Dominant strategy : Strategy that is optimal no matter what an opponent does Equilibrium in dominant strategies: Outcome of a game in which each firm is doing the best it can regardless of what its competitors are doing Not every game has a dominant strategy for each player For player without a dominant strategy, the optimal decision will depend on what the other player does. Nash Equilibrium: A set of strategies (or actions) such that each player is doing the best it can given the actions of its opponents None of the players have incentive to deviate from its Nash strategy, therefore it is stable Example: In the following game, does either player have a dominant strategy? What is/are the Nash equilibrium/equilibria? Firm 1 Sell online Don’t sell online Sell online 50, 60 40, 20 Firm 2 Don’t Sell 0nline 20, 30 60, 40 Answer: Neither player has a dominant strategy and there are two Nash equilibria NE: (Sell online, Sell online) with payoffs of (50, 60) (Don’t sell online, Don’t sell online) with payoffs of (60, 40) 1 Types of games (1) Zero-sum games (constant sum games) The gain of one player equals the loss of other player. The payoffs always add up to zero. Example: Matching pennies Child 2 Child 1 H –1, +1 +1, –1 H T T +1, –1 –1, +1 This is a zero-sum game as the sum of the payoffs of the two players always adds up to 0. Game of simultaneous moves No pure strategy NE in this game (2) Non-zero sum games (variable sum games) Example: Alice Sell Card Bob Sell Card –100, –100 +50, +25 +25, +50 +10, +10 This is a non zero-sum game as both players can get positive or negative payoff, or one positive, one negative at the same time. Game of simultaneous moves Multiple equilibria Number of Nash equilibrium (equilibria) (1) A game can have unique pure strategy equilibrium For example, we would arrive a unique pure strategy equilibrium when its dominance solvable in strict sense. For example, the prisoner’s dilemma. (2) A game with multiple equilibria Example: Matching numbers Player 2 1 2 3 4 5 1 1,1 0, 0 0, 0 0, 0 0, 0 2 0, 0 2, 2 0, 0 0, 0 0, 0 Player 1 3 0, 0 0, 0 3, 3 0, 0 0, 0 4 0, 0 0, 0 0, 0 4, 4 0, 0 5 0, 0 0, 0 0, 0 0, 0 5, 5 We have a game with 5 NEs. In such case, we can apply refinement concepts to our NEs. For example, Pareto dominance. In this example, the Pareto superior outcome is (5,5) out of some notion of rationality. History matters in the sense that an equilibrium say (4, 4), once achieved, tended to perpetuate itself. (3) Games with no pure strategy equilibrium 2 Example: War In this game the two generals have to decide whether to attack or retreat, and the payoffs for different actions is given in the following: General 1 Retreat Attack General 2 Retreat Attack 5, 8 6, 6 8, 0 2, 3 No pure strategy NE. Pure strategy means either to attack or retreat. (with prob. 0 or 1) We can find out mixed strategy NE for the game. Maximin strategy Strategy that maximizes the minimum gain that can be earned A conservative alternative to profit-maximizing strategies – player A’s maximin strategy is the one that guarantees A the best outcome if the other player plays the strategy that is worst for A. Example: Investment decision Two firms with same software standard are considering investing in a new standard. Firm A has a larger market share than Firm B. The payoff matrix is as follows: Firm B Firm A Don’t invest Invest Don’t invest 0, 0 -100, 0 Invest -10, 10 20, 10 Dominant strategy for Firm B is Invest, there is no dominant strategy for Firm A If both Firm A and B are rational, the NE is (Invest, Invest) If Firm B is not rational, and Firm A is conservative, Firm A will choose a strategy that maximizes the minimum gain that can be earned. Firm A will choose its maximin strategy, Don’t invest. (–10 > –100) Similarly, Firm B’s maximin strategy is Invest. Example: Player B Player A Top Bottom Left 1, 1 -1000, 0 Right 0, 2 2, 2 Player B’s dominant strategy is Right. There is no dominant strategy for Player A. NE: (Bottom, Right) If Player A chooses Top, the worst that can happen is that B chooses Right and she gets 0. If Player A chooses bottom, the worst that can happen is that Player B chooses Left and A gets –1000. Top is the maximin strategy – it gets the maximum of the minimum payoffs in each row. Similarly, the maximin strategy for B is to play Right. If both play maximin strategies, (Top, Right) will be the outcome 3 Mixed strategy Nash equilibrium Pure Strategy: Player makes a specific choice or takes a specific action Mixed Strategy: Player makes a random choice among two or more possible actions, based on a set of chosen probabilities Example: War In this game the two generals have to decide whether to attack or retreat, and the payoffs for different actions is given in the following: General 1 Retreat Attack General 2 Retreat Attack 5, 8 6, 6 8, 0 2, 3 No pure strategy NE. Pure strategy means either to attack or retreat. (with probability 0 or 1) We can find out mixed strategy NE for the game. General 1 Retreat (p) Attack (1-p) General 2 Retreat (q) Attack (1-q) 5, 8 6, 6 8, 0 2, 3 Let p be the probability that General 1 chooses Retreat, and (1– p) be the probability of General 1 chooses Attack. Let q be the probability that General 2 chooses Retreat, and (1– q) be the probability of General 2 chooses Attack. A player will be willing to choose a mixed strategy only if the expected payoffs of the pure strategies are equal given the other player’s mixed strategy. Given General 2 mixes his strategy of R and A with probability of q and (1– q), then General 1 will mix his strategy of R and A (indifferent between Rand A) only if 4 5q 6(1 q) 8q 2(1 q) q * 7 5q 6(1 q) is the expected payoff of pure strategy R for G1 given G2’s mixed strategy 8q 2(1 q) is the expected payoff of pure strategy A for G1 given G2’s mixed strategy Given General 1 mixes his strategy of R and A with probability of p and (1– p), then General 2 will mix his strategy of R and A (indifferent between Rand A) only if 3 8 p 0(1 p) 6 p 3(1 p) p * 5 8 p 0(1 p) is the expected payoff of pure strategy R for G2 given G1’s mixed strategy 6 p 3(1 p) is the expected payoff of pure strategy A for G2 given G1’s mixed strategy 3 2 4 3 The mixed strategy NE is p * ,1 p * , q * ,1 q * 5 5 7 7 4 Repeated Games: How to solve the Prisoners’ dilemma? Game in which actions are taken and payoffs received over and over again In the oligopoly price setting game, Cheat is a dominant strategy for both firms. NE: (Cheat, Cheat) with payoff of 500 to each firm. Firm 1 Honor agreement Cheat Firm 2 Honor agreement 500, 500 700, 100 Cheat 100, 700 300, 300 This game is a prisoner dilemma They can both get a higher payoff of 500 by setting High price. Incentive for the two firms to collude They get into agreement to set a higher price in order to have higher profit The collusion is not stable, they will have incentive to set a low price and deviate from the agreement. Suppose the game is a repeated game It is obvious that the gains from continued cooperation may exceed the gains from cheating. Continued cooperation: $500 – $300 for many periods. Cheating: Gain of ($700 – $500) for the period you cheat and Loss of ($500 – $300) for many periods. Ways to make the cooperation stable: If the games are played repeatedly instead of playing once, we can have some ways to sustain collusion. At least we can reduce firms’ incentive to deviate. If the game is a repeated, a firm will have opportunity to penalize the other who cheats. Tit-for-tat: repeated game strategy in which a player responds in kink to an opponent’s previous play, cooperating with cooperative opponents and retaliating against uncooperative ones. (One period punishment) Suppose firm 1 chooses Cheat and firm 2 chooses Honor agreement, then firm 1 will get a higher payoff of 700 and firm 2 gets a payoff of 100. In the next period, firm 2 will penalize firm 1 by choosing strategy “cheat”. Payoff for firm 1 drops from700 to 300. If the game is infinitely repeated, Tit-for-tat strategy is rational If competitor charges low price and undercuts the other firm, it will get high profits that month but know I will lower price next month Both firm will get lower profits if keep undercutting, so not rational to undercut. What if repeat a finite number of times? After the last month, there is no retaliation possible. In the month before last month, knowing that will charge low price in last month, will charge low price in month before By backward induction, we can see that only rational outcome is for both firms to charge low price every month 5 Other solutions: Trigger strategy: a violation is punished for the rest of the repeated game. Suppose firm 1 chooses Cheat and firm 2 chooses Honor agreement, then firm 1 will get a higher payoff of 700 and firm 2 gets a payoff of 100. In the periods afterwards, firm 2 will play “cheat” as to punish firm 1 forever. Self-imposed “most favored customer treatment”: if a product’s price goes down in the future, customers who pay the high prices will be refunded the entire price reduction. Suppose the price difference is $400 normal form under the above practices becomes: Firm 1 Honor agreement Cheat Firm 2 Honor agreement 500, 500 300, 100 Cheat 100, 300 300, 300 In this game, both firms will have better incentive to honoring the agreement, as the action of cutting price is less profitable. Threats, commitments, credibility and entry deterrence Strategic Move: action that gives a player an advantage by constraining his behavior Credible threat, incredible threat (empty threat), reputation, commitment Example: Two office supply stores, Office Emporium and Office Station compete in a local market. Office Emporium is planning to cut prices storewide in order to gain market share. Office Station has stated in a press release that it will cut prices by even more in order to maintain its position in the market. The payoff matrix below shows profits for the two office supply stores Office Station Maintain price Lower price Office Emporium Maintain price Lower price 20, 14 16, 20 4, 1 2, 4 Is the treat of a potential low price campaign by Office Station credible? Obviously, Office Station will be better off maintaining prices if Office Emporium lowers its prices. The threat by Office Station is not credible What if the game is played repeatedly? 6 Example: The entry game and entry deterrence Potential Entrant Enter Don’t Enter Tough -1, -2 0, 10 Incumbent Accommodate 1, 2 0, 10 There are two Nash Equilibria: NE: (Enter, Accommodate) with payoffs (1, 2) (Not enter, Tough) with payoffs (0, 10) Is the threat of being tough by the incumbent credible? The threat of acting tough is not credible if the incumbent has no way of committing to a “tough” strategy, then the potential entrant should enter market. Once the potential entrant is in the market, it will be in the incumbent’s best interest to accommodate How could the incumbent make its threat credible? The incumbent could make some irrevocable commitment action to convince the potential entrant that entry will be unprofitable These commitments will change the incumbent’s payoffs Examples: signing long term contracts to engage advertising campaign, expand capacity prior to entry Potential Entrant Enter Don’t Enter Tough -1, 1 0, 10 Incumbent Accommodate 1, 0.5 0, 10 Costs are incurred prior to entry, but the strategy can be carried out with more planning which lower the costs of acting tough. If the potential entrant enters, the incumbent will act “tough” (1> 0.5) The threat is credible 7
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