1. Week 3 Dominance and Iterated Dominance We define a strategy for a player, the sets of strategies available to a player for a game, and the strategic form. Then we explain the meaning and application of a dominant strategy and iterated dominance.

2. How much do you need to know about the other players? In the cola war, if Coke never acquiesces to Pepsi, then the best response of Pepsi is to enter Canada, but if Coke only responds in kind in Canada, then Pepsi should enter the African American market. Similarly in the Ware case, the best response of each firm depended upon the probability that the other firm would seek to develop the patent. Under what conditions does your best response depend on the choices of the other players?

3. Strategies How general is the idea that the best response may not depend on what other players are doing? To answer this question we introduce and define the concept of 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.

4. Another way of representing games Rather than describe a game by its extensive form, one can describe its strategic form. The set of a player’s (pure) strategies is called the strategy space. The strategic form of the game is a list of all the possible pure strategies for each player 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. Note that no information is lost when transforming a simultaneous move game from its extensive to its strategic form.

5. Strategic form To summarize: the strategy space is the set of mutually exclusive collectively exhaustive (MECE) strategies. 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.

6. Simultaneous move games In many situations, you must decide all your moves without knowing what your rivals are doing, and their situations are similar to yours. Even if the moves are not literally taking place at the same moment, but all the moves must be made before anybody can react, the moves are effectively simultaneous. 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 Ware case is an example of a simultaneous move game.

7. Acquiring Federated Department Stores 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.

8. 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).

9. 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.

10. The game between shareholders After the offers are made, Federated shareholders play an acceptance/rejection game. Each shareholder asks what proportion of their shares should be: 1. sold to Macy’s 2. sold to Campeau’s 3. retained. Note the payoffs received by each shareholder depend on what the other shareholders do.

11. The payoff matrix to a stockholder Campeau succeeds Macy’s succeeds Both fail Tender to Campeau90+15(50/x)105 Tender to Macy’s90102100 Do not tender90 100 Shareholders are better off as a group tendering to Macy’s, but each individual shareholder is better off tendering to Campeau, regardless of what the other players do.

12. Dominant strategies Strategies that are optimal for a player regardless of what the other players do are called 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. Similarly when a dominant strategy exists, the player does not need to know the behavior of the other players to form his or her best response to the probability distribution characterizing their choices.

13. Rule 3 A dominant response is always a best response to the empirical distribution. Therefore if you have a dominant response, you should play it.

14. Investment broker This game is neither a simultaneous move game, nor a perfect information game. However the second mover, the client, knows more than the first mover, her broker.

15. Solution to investment broker The broker should choose “tech” because it is a dominant strategy. If the investor recognized that the broker had a dominant strategy, the investor would use the signal she receives about the economy, picking the strategy “continue if new, liquidate if bubble”. In that case she would be using the principle of iterated dominance to solve the game.

16. Predators This game features a corrupt government, a poorly run state enterprise and an opportunistic foreign investor wrestle for mineral and oil wealth.

17. Government will seize assets if given the opportunity A dominant strategy for the government in this game is to seize foreign assets when presented with the opportunity to do so.

18. The reduced game upon anticipating the government’s choice We now reduce the game by conditioning on the government’s dominant strategy. If the government seizes its assets, the foreign firm should withdraw if it can. If its assets were left intact, it should commit.

19. A further reduction of the predator game Folding back the final decisions of the foreign investor, the further reduction yields a simultaneous move game. Note the state owned enterprise has a dominant strategy to propose a joint venture.

20. Just in time 3/20*50+17/20*50=50 3/20*400-17/20*60=9 3/20*2+17/20*2=2 3/20*248-17/20*12=27

21. Reduced game for component supplier Taking the expected value over the nodes that involve nature (and the possibility of breakdown), we obtain the reduced game depicted on the right. The subgame reached by Pratt & Whitney choosing “Wait” and Boeing choosing “Wants part” is itself a perfect information game, and trivially solved by the choice “Make part”.

22. Strategic form for the reduced game Folding the solution of the subgame into the extensive form, we see the resulting is a 2 by 2 simultaneous move game with the strategic form depicted.

23. MBA market CMU, Pitt and Duquesne compete in their MBA evening programs, drawing from an overlapping demand pool. Their reputations and cross synergies with other programs effectively shape the kinds of choices they offer. One of the players has a dominant strategy, and the game can be solved using iterative dominance.

24. Dominated strategies If a player has a dominant strategy, then her other strategies are called dominated. More generally, a strategy is called dominated if there exists some other strategy yielding a higher expected payoffs regardless of the strategies that the other players pick. Thus a person should never play a dominated strategy. She can earn more by choosing another strategy without knowing the choices of the other players in the game.

25. Marketing groceries In this simultaneous move game the corner store franchise would suffer greatly if it competed on the same feature as the supermarket. This is illustrated by the fact that its smallest payoffs lie down the diagonal.

26. Strategies dominated by a mixture The supermarket's hours strategy is dominated by a mixture of the price and service strategies. Let π denote the probability that the supermarket chooses a price strategy, and (1-π) denote the probability that the supermarket chooses a service strategy. This mixture dominates the hours strategy if the following three conditions are satisfied: π65+(1-π)50 > 45or π > -1/3 π50+(1-π)55 > 52or 3/5 > π π60+(1-π)50 > 55or π > ½ Hence all mixtures of π satisfying the inequalities: ½ < π < 3/5 dominate the hours strategy.

27. Revisiting “look ahead and reason back” Let us reconsider the first rule we derived of “look ahead and reason back” for games of perfect information. At the bottom node aren’t we just eliminating those strategies that are dominated by any strategy that picks out the best move at the very end of the game? This insight leads us to the following theorem, which can be easily proved by an induction: Games of perfect information can be solved by writing down their strategic form and applying Rule 3. So let us rewrite Rule 3 in terms of iterated dominance:

28. Rule 3 (revised) All players should simplify the game by iteratively discarding dominated strategies.

29. Three rules for strategic play Rule 1:Play your best response to the population distribution of conditional choiceprobabilities. Rule 2:If there is a dominant strategy, play it. Rule 3:Iteratively eliminate dominated strategies. (Note that Rule 3 covers look ahead and reason back.)

30. How sophisticated are the players? Applying the principle of iterative dominance assumes players are more sophisticated than applying the principle of dominance. Applying the dominance principle in simultaneous move games makes sense as a unilateral strategy. In contrast, a player who follows the principle of iterative dominance does so because he believes the other players choose according to that principle too. Each player must recognize all the dominated strategies of every player, reduce the strategy space of every player as called for, and then repeat the process.

31. Why is dominance more compelling than iterated dominance? If you don’t know the choice probabilities of the other players, the two most important principles for strategic play, are to play dominant strategies, and iteratively eliminate dominated strategies. The first principle applies regardless of whether the other players are rational or not, and therefore does not depend on whether you know their payoffs or not. The second principle applies when you know enough about the payoffs of the other players to recognize their dominated strategies.

32. Summary We developed the strategic form of a game, and demonstrated that some games are easier to analyze in their strategic form than in their extensive form. We analyzed two further principles for strategic play, based on dominance, and iterated dominance. Sometimes a particular strategy gives the highest payoffs regardless of what the other players do. You don’t have to know anything about their empirical strategies or their payoffs you have a dominant strategy to play. Iterative dominance is more subtle; it encompasses backwards induction as an application. Here you bank on everyone being smart enough not to play a dominated strategy, and everyone being is as smart as you.