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TOPIC 5 SEQUENTIAL MOVE GAMES WITH PERFECT INFORMATION. Sequential games: Strategic interactions where there is a strict order of play. At least one player has some information about his opponents´ choices when he has to make a decision. He knows part of the previous history of the play of the game. PERFECT INFORMATION (perfect observability): players know everything that has happened prior to making a decision. Example: chess, checkers…

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Topic 5 : GAME TREES We represent sequential games in extensive form, which means: - a complete and detailed list of all possible complete plays (paths, histories) of the game (sequences of actions from the beginning until the end of the game) and, - the final payoffs associated with each of these paths. A way to capture all these elements is a GAME TREE. - Nodes (or decision nodes): situations in which a player has to make a decision. - Branches: feasible actions in each node. - Terminal nodes: final outcomes or payoffs.

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ENTRY DETERRENCE A monopolist incumbent firm (M) enjoys monopoly profits of 2 million euros in a market A potential competitor (the Entrant, E) makes a decision between entering in the market (action I) or staying out (action O). The monopolist has to choose, in case of entry, between accomodating, sharing the monopoly profits with E (action A), or cutting prices below costs so that neither you nor your competitor could make profits (action F, for fighting). In this latter case, both firms would obtain losses of 1 million each.

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THE STREET GARDEN GAME Two players live on the same small street. Each has been asked to contribute with 2 monetary units toward the creation of a flower garden. The ultimate quality of the garden depends on how many of them contribute. If both contribute, they get a pleasant garden which yields an individual utility of 4 monetary units to each player. If only one of them contributes, the garden yields 3 monetary units to each player. And, if no player contributes, there will be no garden (utility zero). Suppose the players move sequentially in a previously established order. That is, player 1 has the first move, and chooses whether to contribute. Then, after observing what player 1 has chosen, player 2 makes her choice.

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STRATEGIES for each player are complete plans that describe actions at each of the player´s decision nodes contingent on all possible combinations of actions made by players who acted at earlier nodes. In general, they are complete plans of action contingent on the information available at each moment of the game.

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PERFECT NASH EQUILIBRIUM. A way to rule out Nash equilibria that represent non-credible strategic moves (threats, promises…) is to require the principle of sequential rationality to be satisfied. The principle of Sequential Rationality (SR) says that all players, when constructing their strategies in any node, should anticipate future rational behaviour of their opponents. A Nash equilibrium which satisfies the principle of SR is called a perfect Nash equilibrium (PE). In finite sequential games (with a finite number of turns) the PE are computed by BACKWARD INDUCTION (or rollback), that is, solving the game from the end until the initial node. Players decide their current moves on the basis of calculations of future consequences. All finite sequential game with perfect information has a PE and “almost always” is unique.

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A PUCCINI OPERA In Puccini´s opera Gianni Schicchi, Buoso Donati has died and left his large estate to a monastery. Before the will is read by anyone else, the relatives call in a famous mimic, Gianni Schicchi, to play Buoso on his deathbed, re-write the will, and then convincingly die. Schicchi explains very carefully to the relatives, how severe are the penalties for cheating with a will (at the time, it included having one´s hand cut off). The plan is put into effect. On the deathbed, Schicchi, as Buoso Donati, rewrites the will leaving the entire estate… to the famous mimic and great artist, Gianni Schicchi.

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THE STREET GARDEN GAME Suppose that player 2 publicly announces a contract with a third player, the delegate (d), before the garden game takes place. The contract states that the game will be played by player d, instead of player 2, and that they will share equally the net gains obtained in the game. It is common knowledge between players 1 and 2 that player d is inequity averse. In particular, his utility function, given material payoffs x 1 and x d, is given by: U d = x d – 2.max{x 1 – x d, 0}.

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STRATEGIC MOVES. Strategic moves are devices to manipulate the rules of the game: a strategic move as the 1st stage and the altered original game as the 2d stage. Three types of strategic moves: commitments, threats and promises. The aim of all three is to alter the outcome of the 2d stage to your own advantage. All require CREDIBILITY: the other player must believe that you will not renege, that you will follow through. Mere declarations of intentions are not enough. You need extra moves in the 1st stage to make them credible.

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STRATEGIC MOVES - A commitment or unconditional move is a (response) rule in which you move first and your strategy is fixed. Intended to gain first-mover advantage. - Threats and Promises occur when you move second: they are response rules. Different from the best response in the original game, conditional on what the other side does. They should be communicated before the other player moves. Intended to gain second-mover advantage.

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SEQUENTIAL INSPECTION An employee can shirk (S) at a zero cost or work hard (W) incurring in a cost of 6. Shirking yields zero output for the employer, while working hard yields an output of 16. The employer can make an inspection with a cost of 4, that will provide evidence about the employee´s behaviour. The contract estipulates a wage of 8 to be payed unless there is evidence on shirking (that is, it is not possible to condition the wage on output). If the inspection finds shirking, the employee is fired (zero payoff). A) Suppose the game is played sequentially: the employer moves first and the employee after observing his decision moves second. Find the perfect Nash equilibrium. B) Now, the employer announces a probability of inspection. The employee chooses his action after observing the announcement. Finally, there is an inspection or not, according to the realization of the announced lottery. Find the perfect equilibrium of this new game. What problems would appear if the lottery is not public?

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PROMISES A mother makes a verbal promise to his son: if you are good boy tomorrow, I will buy you a present. The son makes a decision between behaving well or wrongly and, in case he has been a good boy, the mother chooses between fulfilling her promise or not. A consumer chooses whether or not to purchase a service from a firm. If the consumer does not purchase, then both players receive a payoff of 0. If the consumer decides to purchase, then the firm must decide whether to produce high or low quality. In the former case, both players have a payoff of 1. In the latter case, the firm´s payoff is 2 and the consumer´s payoff is -1.

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HOW TO ACHIEVE CREDIBILITY Two general principles to achieve credibility of a strategic move: -A- Limiting your strategic freedom, in a way that you have no other option but to follow through the strategic move. -B- Changing your future payoffs, in a way that to follow through the strategic move is payoff- maximizing for you. Practical methods to implement these ideas:

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HOW TO ACHIEVE CREDIBILITY -A- Limiting your strategic freedom. 1- Leaving the outcome beyond your control. 2- Delegation: mandated negotiating agent with appropiate incentives that should be different from yours. 3- Burning your bridges or sinking your ships (eliminate actions). 4- Cutting off communication.

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HOW TO ACHIEVE CREDIBILITY -B- Changing your future payoffs. 1- Reputation: if you play repeatedly with the same or different opponents, it might be worthwhile to invest in a reputation of fulfilling always your threats and promises. 2- Moving in small steps: break the threat or promise into many small pieces. Ex: Paying the constructor of your house. 3- Teamwork (peer pressure). 4- Rational irrationality (it makes sense in an incomplete information scenario). 5- Contracts (but beware, contracts can be renegotiated).

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FRIGHTENING MANY WITH ONE THREAT Assume that you have ten employees. Each one gets a payoff of 5000 euros from working hard, a payoff of 10000 euros from shirking and a payoff of 0 from getting fired. Effort is verifiable and you want all them to work hard. But, you can fire, at most, one employee, and your employees know this. How can you make them all work hard?

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CHANGING THE ORDER OF MOVES What would happen in the simple simultaneous games we studied in previous lectures, if players move sequentially and with perfect information? Does the outcome of the game change? Does it matter to move first or second? Does any player improve (or not) his payoff? Can both players do better in the sequential game? We will check this possibility for the Prisoners´ Dilemma, the Assurance game, the Chicken game and other strategic structures.

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ULTIMATUM GAME A seller and a buyer bargain on the price of an indivisible good. The monetary valuation of the buyer is 140 euros and the seller´s provision cost is 130 euros. This is common knowledge. The seller announces a price and the buyer accepts or rejects it. In case of rejection, there is no trade. Two individuals can share 10 euros but only after agreeing how to divide them. The rules of the negotiation are: player 1 (the proposer) makes a take-it-or-leave-it offer of division and player 2 (the responder) accepts or rejects it. If there is rejection the game ends without an agreement.

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ULTIMATUM GAME: experimental regularities. 1. There are virtually no offers above 0.5. 2. The vast majority of offers in almost any study is in the interval [0.4, 0.5]. 3. There are almost no offers below 0.2. 4. Low offers are frequently rejected and the probability of rejection decreases in general with the size of the offer. 5. If the responder´s option to reject is removed (the dictator game), the proposers become significantly less generous.

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