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Lecture 5The Strategic Form This lecture provides a second way of representing games, via its strategic form. We define a strategy for a player, and the sets of strategies available to a player for a game. The set of all strategies for every player and all payoffs that can arise from any permutation of strategies defines the strategic form. We show how to derive the strategic form from the extensive form and demonstrate with examples why it is sometimes more useful to analyze the strategic rather than the extensive form.
The three elements of the strategic form • Up until now we have defined every game in its extensive form, with the four elements: • Players • Actions • Information • Payoffs • There is another way of representing any game, that is using its strategic form. This has three elements: • Players • Strategies • Expected payoffs. • Some games are easier to solve, and its outcomes of easier to predict, if we analyze its strategic form, rather than its extensive form.
Strategies • At the heart of the definition of the strategic form is 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. • To understand this concept and appreciate its usefulness, we now consider some examples.
Revisiting Coke and Pepsi • Recall the extensive form of the Coke-Pepsi game looks like: • We now derive its strategic form.
The Strategy Space for Pepsi • In this game Pepsi is a first mover, and makes no further moves in the game. • Furthermore nature (consumer demand tastes) follow Pepsi’s move (or equivalently Pepsi moves without knowing nature’s moves. • Therefore each of Pepsi’s moves corresponds to precisely one strategy. • Thus the Pepsi has are two strategies: • Advertise in Quebec. • Advertise to African Americans.
The Strategy Space for Coke • Like Pepsi Coke has only one move to make this game. • However Coke moves second, and can see the move Pepsi has taken before it makes its own. • Therefore Coke can condition on what Pepsis does in formulating its own strategy. • Thus Coke has four strategies: • Always acquiesce. • Always respond in kind. • Acquiesce if Pepsi advertises in Quebec and respond in kind if Pepsi advertises to African Americans. • Acquiesce if Pepsi advertises to African Americans and respond in kind if Pepsi advertises in Quebec.
Matching Strategy Pairs to Expected Payoffs • For each strategy pair we can calculate the expected payoff to both of the players. • Suppose Pepsi advertises in Quebec and Coke always acquiesces. That is they both choose their first strategy. • Then Pepsi’s payoff is 6 and Coke’s is -3. • Alternatively if both Pepsi and Coke choose their second strategy, Pepsi advertising to African Americans and Coke responding in kind, then the expected payoff to: • Pepsi is -10*0.4 + 4*0.6 = -1.6 • Coke is -30*0.4 + 3*0.6 = -10.2
The Strategic Form Illustrated • The bi-matrix is created by forming rows from Pepsi’s strategies, columns from Coke’s strategies and filling in the expected payoffs in the cell corresponding to each strategy pair.
Comparing the solutions of the strategic and extensive forms • Coke playing the far right strategy, acquiescing only if Pepsi advertises to African Americans (the fourth strategy), guarantees itself the highest payoff regardless of what Pepsi does. • For this reason Coke’s fourth strategy is called a (weakly) dominant strategy. • If Coke plays this strategy, then the best strategy of Pepsi is to advertise to African Americans. • This is exactly the solution we found by applying the backwards induction to the extensive form!
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 are 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. In simultaneous move games the strategic and extensive forms essentially represent the same details.
A simultaneous move game between Pepsi and Coke • Connecting a dotted line to the two decision nodes for Coke, prevents Coke from seeing which market Pepsi has advertised in. • The extensive form for this new game cannot be solved using backwards induction.
The strategies for Pepsi and Coke in this simultaneous move game • As in the perfect information game Pepsi has just two strategies: • Advertise in Quebec. • Advertise to African Americans. • In contrast to the perfect information game Coke cannot condition on what Pepsi has done in this game. • It is as if they moved simultaneously: hence the name. • Thus Coke also only has two strategies: • Respond in kind. • Acquiesce.
Pairing the Strategies with the Expected Payoffs in this Simultaneous Move Game • As in the perfect information game Pepsi has just two strategies: • Advertise in Quebec. • Advertise to African Americans. • In contrast to the perfect information game Coke cannot condition on what Pepsi has done in this game. • It is as if they moved simultaneously: hence the name. • Thus Coke also only has two strategies: • Respond in kind. • Acquiesce.
The strategic form of the simultaneous move game • Thus the strategic form of this game is found by simply lopping off the last two columns of the original game:
Solution to the Game • Regardless of which Coke picks the best strategy for Pepsi is to advertise in Quebec because 6 > 0.8 and -1 > -1.6. • This is called a (strictly) dominant strategy. • Coke can do this calculation just as easily as Pepsi. Consequently Coke anticipates Pepsi will advertise in Quebec. • Coke responds by choosing to respond in kind because -2 > -3.
Comparing the solutions • This is exactly the opposite to the solution we obtained for the perfect information game. • This Pepsi increase its payoff by 1.8 from a loss to a profit if it can convince Coke where it has advertised before Coke makes a decision. • This is called a first mover advantage.
Investment broker • In this game a decides whether to invest in tech or industrial stocks. • The client does not see which stocks the broker invests in, but does see whether she is living in a new economy or a bubble. • Thus she knows more than the broker.
Strategies for both players • The broker has two strategies: • Tech. • Industrial. • The client however has four strategies: • Always continue with broker. • Always liquidate. • Continue with broker if she is living in a bubble and liquidate if she is living in a new economy. • Continue with broker if she is living in a new economy and liquidate if she is living in a bubble.
Computing the expected payoffsfor the strategy pairs • Consider the following strategy pair: • The broker chooses “tech” • The client chooses “Continue with broker if she is living in a bubble and liquidate if she is living in a new economy. • What is the expected payoffs to each player? • From the game tree we see that the: • broker gets 0.5(3 + 4) = 3.5. • client gets 0.5(-9 + 2) = -3.5. • The expected payoffs for the other strategy pairs are calculated in exactly the same way.
Strategic form of this game • For example, suppose the broker chooses “tech” and the client’s strategy is “continue with broker”. The broker’s expected compensation is: 0.5*3 + 0.5*9 = 6 • Let the rows denote the strategies of the client and the columns denote the strategies for the broker. • Then fill in the expected payoffs for each strategy pair, in the corresponding cell.
Solution to investment broker • The broker should choose “tech” because regardless of which strategy • “Tech is an example of 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.
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. • Not much information is lost when transforming a simultaneous move game from its extensive to its strategic form. But different extensive forms have the same strategic form.
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.
Summary • The strategic form is complementry to the extensive form. • The extensive form of a game is based on: • Players • Actions • Information • Payoffs • The strategic form of the same game is based on: • Players • Strategies • Expected Payoffs • We showed: • how to derive the strategic form from the extensive form. • that some games are easier to solve using the strategic form rather than the extensive form.
