THE PROBLEM OF MULTIPLE EQUILIBRIA NE is not enough by itself and must be supplemented by some other consideration that selects the one equilibrium with better claims to be the prediction. Two approaches: - to analyze the learning population dynamics: does it converge to any equilibrium? - to require an additional “reasonable” property (besides the equilibrium property). FOCAL POINT (T. Schelling): “something” in the structure of the game or in the common social, cultural, or historical knowledge of all the players enables their expectations to converge to a particular equilibrium.

PURE COORDINATION GAMES. - Common interests, but independent choices. Examples: “we had a flat tire”, to coordinate in a letter from the alphabet, the traffic game. - With previous communication, there is no strategic problem. But, without communication, you need a focal point.

ASSURANCE OR TRUST GAMES. These are coordination games where there is a conflict between efficiency and risk. There is always a safe strategy and a risky strategy for each player (but the risky strategy yields the efficient outcome if everybody plays it). RISK DOMINANCE. Suppose a two-player game with two actions for each player and where there are two NE. The risk dominant NE is the equilibrium in which the players choose the action that is their best response if they expect that his opponent will play both actions with equal probability. Example: Incentives in a team where levels of effort are complementary inputs.

A GAME OF SIMULTANEOUS DEMANDS. Two players can divide between them 100 euros, but only in case they reach an agreement. They have to write down in a closed envelope their demands, that is, the amount of euros they want for themselves. Let us denote the demands as x 1 and x 2. You can only write integers and 0 and 100 are not allowed. If the demands are compatible, x 1 + x 2 ≤ 100, each player obtains his own demand. But if they are incompatible, x 1 + x 2 > 100, both players get a zero payoff. Find the Nash equilibria of this game.

A firm assigns two employees to complete a task. They will make a simultaneous decision between high effort (e = 2) or low effort (e = 1). The total revenue generated by the team is given by the function R = 6.(e 1 + e 2 ) and the individual cost of effort is for both players c(e i ) = 4e i, i = 1,2. The firm and each worker sign contracts that estipulate that the workers will share equally the obtained revenue less two monetary units for the firm, only if the revenue is greater or equal than 18. If the revenue is smaller than 18, both workers would be fired (a payoff of zero) (because this constitutes objective evidence in a court that none of them has exerted high effort). Find the NE of this game.

CHICKEN GAME Two teenagers take their cars to opposite ends of a road and start to drive very fast toward each other. The one who keeps going straight is the winner and the one who swerves to prevent a collision is the “chicken”. Four essential features: 1) Each player has one strategy that is the “tough” strategy and one that is the “weak” strategy. 2) There are two NE. These are the outcomes in which exactly one of the players is weak. 3) Each player strictly prefers that equilibrium in which the other player chooses weak. 4) The payoffs when both players are tough are very bad for both players. The real game becomes a test of how to achieve one´s preferred equilibrium by convincing your rival that you are going to play tough.

More chicken games: - Innovate or imitate (to imitate is less costly). - Two firms consider to enter a local market in which there is room for only one firm to operate profitably. - Multiple firms conducting similar research. The first one to make the discovery, receives the patent rights. - The hawk – dove game. Two individuals of the same species compete for a resource. The Hawk strategy (H) is aggresive and fights to try to get the whole resource of value V. The Dove strategy (D) is to offer to share but to shirk from a fight. When two H types meet each other, they fight. Each animal is equally likely to win and get V or to lose, be injured, and get –C. When two D types meet, they share without a fight. When a H type meets a D type, the latter retreats.

TOPIC 4 SIMULTANEOUS GAMES WITH INCOMPLETE INFORMATION. Some pieces of relevant information are known by some players (private information) but not by others before the game is played. That is, some players do not know the true payoffs or motivation of their opponents. Examples: - a seller knows the actual quality of the good while the buyer does not. - a recently hired worker knows his true productivity while the firm does not. - A company may not know the true cost function of its competitor. - A buyer in an auction may not know his oponents´ true valuation of the object. - You may not know if you are playing against an opponent with selfish preferences or altruist or inequity averse preferences.

BAYESIAN GAMES The player with incomplete information knows all the possible specifications of his rivals´ private information (the types of the other players) and its objective probabilities. Moreover, this is common knowledge among the players. He does not know the true type of his opponent but he knows the population where his opponent comes from. - This is a BAYESIAN GAME (Harsanyi): players, sets of actions, sets of types, a probability distribution on the set of types and the payoff functions of the players (which will depend on actions and types).

INNOVATE OR IMITATE. Two companies are involved in the following simultaneous game. Any of them can develop a product innovation with a cost of 1 million euros, but its competitor can imitate instantaneously and with zero cost. The total expected profits of the innovation will be 4 million which will be shared equally by the firms. Firm 1 has these costs of innovating and this is common knowledge but the true cost of innovating of firm 2 are private information and can be 1 million with probability 0,6 or 3 million with probability 0,4. Find the NE of this game.

Strategies and actions are different concepts for players with private information (with types). A strategy of a player is a complete plan that specifies one action for any of his possible types (and not only for his true type). A Bayesian Nash equilibrium (BNE) in an incomplete information two-player game is a pair of strategies, one for each player, which are mutually best- responses. That is, the action specified in the equilibrium strategy for each player´s type has to be a best response to the other player´s strategy.

MORE INFORMATION CAN HURT YOU The following game is a prisoners´ dilemma for player 1 and a coordination game for player 2. C denotes the cooperative action and NC the non-cooperative action. However, player 1 has also a more “destructive” non- cooperative action. Notice that it is a dominated action for player 1. Assume that it is private information of player 2 which is the true destructive NC action. Player 1 assigns ½ probability to each possibility. Find the BNE and compare the result with the complete information case. CNC C6,104,8 NC18,45,6 NC20,02,0

A FIRST-PRICE SEALED-BID AUCTION. Suppose a first-price sealed- bid auction between two buyers. The monetary valuation of player 1 is 100 and it is common knowledge. But the valuation of player 2 is private information and it can take two values: 50 with probability 4/5 and 75 with probability 1/5. Show that the result of any BNE is inefficient.