Lecture 2A Strategic form games Not every game can be solved using the extensive form. This lecture shows how the strategic form can be derived from the extensive form. Then we show how simply the solution of a game is derived if each player has a dominant strategy. Readings: Chapter 7 of Strategic Play

Imperfect Information Every game that does not have perfect information is called a game with imperfect information. Games with imperfect information cannot be solved using backwards induction alone, because there are nodes at which the player designated to make a choice cannot observe everything that has transpired in the game. These nodes invariably belong to information sets that contain more than one node.

Overview In many situations, you must determine your strategy without knowing what your rival is doing at the same time. Even if the moves are not literally taking place at the same moment, but both moves are made in ignorance of the rival’s, the moves are effectively simultaneous.

Expansion

Large producer moves first

Small producer moves first

The limits of backwards induction The principle of backwards induction cannot be applied in simultaneous move games, and is of limited use in solving games with imperfect information. For these reasons we introduce another representation of a game that does not exploit the order of play at all, called the strategic form, and we will seek solutions to imperfect information games using this alternative representation.

A second way of representing games Rather than describe a game by its extensive form, one can describe its strategic form. The strategic form of the game is a list of all the possible pure strategies for each of the players and the (expected) payoffs resulting from them. Suppose every player chooses a pure strategy, and that nature does not play any role in the game. In that case, the strategy profile would yield a unique terminal node and thus map into payoffs.

Strategies The foundation of the strategic form is a strategy. A strategy is a full set of instructions to a player, telling her how to move at all the decision nodes assigned to her. Strategies respect information sets: the set of possible instructions at decision nodes belonging to the same information set must be identical. Strategies are exhaustive: they include directions about moves the player should make should she reach any of her assigned nodes. The set of a player’s strategies is called the strategy space.

Strategic form The strategic form representation is less comprehensive than the extensive form, discarding detail about the order in which moves are taken. The strategic form defines a game by the set of strategies available to all the players and the payoffs induced by them. In two player games, a matrix shows the payoffs as a mapping of the strategies of each player. Each row (column) of the table corresponds to a pure strategy. The cells of the table respectively depict the payoffs for the row and column player.

Simultaneous move games A game where no player can make a choice that depends on the moves of the other players is called a simultaneous move game. The strategic form of simultaneous move games has special significance, because in contrast to all other games, no information is lost when transforming a simultaneous move games from its extensive form to its strategic form.

Strategies in the simultaneous move game of expansion In simultaneous games, strategies for each player correspond to the moves themselves. That is because each player has only one information set. In the simultaneous move game of expansion for example, there are only two strategies for each player, and in this game the set of strategies for the other player in this game is the same: - to expand - to maintain the current capacity.

Strategic form of the simultaneous move expansion game

Strategies for small producer when large producer moves first 1. Always increase capacity. 2. Always retain current capacity. 3. Do the same thing as the large producer. 4. Do the opposite of what the large producer does.

Strategic form of expansion game where large producer is first mover Comparing this matrix with the matrix depicting the simultaneous move game, we see they are not the same. The differences are directly attributable to the different information sets.

An example of a dominant strategy The small producer cannot achieve a higher payoff by deviating from the strategy of “do the opposite to large producer”. This is called a dominant strategy.

Acquiring Federated Department Stores Here is another example of a simultaneous move game. Robert Campeau and Macy's are competing for control of Federated Department Stores in 1988. If both offers fail, then the market price will be benchmarked at 100. If one succeeds, then any shares not tendered to the winner will be bought from the current owner for 90. The argument here is that losing minority shareholders will get burned by the new majority shareholders.

Campeau’s offer . . . Campeau made an unconditional two tier offer. The price paid per share would depend on what fraction of the company Campeau was offered. If Campeau got less than half, it would pay 105 per share. If it got more than half, it would pay 105 on the first half of the company, and 90 on any remaining shares. Each share tendered would receive a blend of these two prices so that every share received the average price paid. If a percentage x > 50 of the company is tendered, then 50/x of them get 105, and (1 - 50/x) of them get 90 for a blended price of: 105* 50/x + 90(1 - 50/x) = 90 + 15(50/x).

Macy’s offer . . . Macy's offer was conditional at a price of 102 per share: it offered to pay 102 for each share tendered, but only if at least 50% of the shares were tendered to it. Note that if everyone tenders to Macy's, they receive 102 per share, while if everyone tenders to Campeau, they receive 97.50. so, shareholders are collectively better off tendering to Macy's than to Campeau.

The payoff matrix to a stockholder Although share holders are better off as a group tendering to Macy’s, each individual shareholder is better off tendering to Campeau, because it is a dominant strategy. Campeau succeeds Macy’s succeeds Both fail Tender to Campeau 90+15(50/x) 105 Tender to Macy’s 90 102 100 Do not tender

Strictly dominant strategies Strategies that are optimal for a player regardless of whether the other players play rationally or not are called dominant. If a dominant strategy is unique, it is called strictly dominant. Although a player's payoff might depend on the choices of the other players, when a dominant strategy exists, the player has no reason to introspect about the objectives of the other players in order to make his own decision.

Prisoner’s dilemma The solution is that both Defect, which is Pareto inferior to the outcome of both Cooperate. The interactive reasoning here is based on strictly dominant strategies for both players. The prisoner’s dilemma shows that when individuals act rationally, the outcome for the everyone in the group is not necessarily as good as it could have been.

Prisoner’s dilemma situations Cooperate Defect Oligopoly: low output high output Teams : high effort low effort Joint Research: reveal your hide your knowledge knowledge Common property: careful use exploitative use

Fallacy of group rationality The paradox of the prisoners’ dilemma, and examples of perfect information games we played, illustrate the folly of believing that groups have an innate propensity to collectively and reflexively implement its members’ common goals. Those examples show that individual rationality is instrumental in undermining common goals. Incentives within an organization determine group behavior, a recurring theme of 45-976: Bargaining, Contracts and Strategic Investment.

If you have a dominant strategy, use it. Rule 2 If you have a dominant strategy, use it.

Lecture summary The backwards induction principle derived for perfect information games cannot be easily extended to games of imperfect information. We defined the strategic form of a game. Some games are easier to analyze when presented in their strategic form than in extensive form. In particular, our second rule for good strategic play, playing a dominant strategy where possible, is applied to the strategic form.