1. A camper awakens to the growl of a hungry bear and sees his friend putting on a pair of running shoes, “You can’t outrun a bear,” scoffs the camper. His friend coolly replies, “ I don’t have to. I only have to outrun you!”

2.  A game is any competition between players (such as individuals or firms) in which strategic behavior plays a major role.  An action is a move that a player makes at a specified stage of a game, such as how much output a firm produces in the current period.  For example, a firm may use a simple business strategy where it produces 100 units of output regardless of what any rival does.

3.  Or the firm may choose a more complex strategy in which it produces a small quantity as long as its rival produced a small amount last period, and a large quantity otherwise.  The payoffs of a game are the player’s valuation of the outcome of the game, such as profits for firms or utilities for individuals.

4.  Strategic behavior is a set of actions a player takes to increase his or her payoff, taking into account the possible actions of other players.  For example, a firm may set an output level, act to discourage potential firms from entering a market, or choose to employ a technology.  Conflicts frequently arise among firms because the actions of each profit- maximizing firm affect the profits of other firms.

5.  Assumptions: 1.Players are interested in maximizing their payoffs. 2.Players have common knowledge about the rules of the game, that each player’s payoff depends on actions taken by all players, and that all players want to maximize their payoffs; all players know that all players know the payoffs and that their opponents are payoff maximizing; and so on.  Economists use game theory when a player’s optimal strategy depends on the actions of others, which is called strategic interdependence.

6.  A game is described in terms of the players; its rules; the outcome; the payoffs to players corresponding to each possible outcome; and the information that players have about their rival’s moves.  The rules of the game determine the timing of players’ moves and the actions that players can make at each move.

7.  For each game, a payoff function determines any player’s payoff given the combination of actions by all players.  Games with complete information are games where the payoff function is common knowledge among all players.  Each player knows the payoffs to all the players in the game for any possible combination of strategies.

8.  Game theorists distinguish between complete information and perfect information, where the player who is about to move knows the full history of the play of the game to this point, and that information is updated with each subsequent action.  The information each player had about rivals, actions often turns on whether the players act simultaneously or sequentially.  If the players move simultaneously, then they have imperfect information because they do not know how other players will act.

9.  In static games, the players choose their actions simultaneously, have complete information about the payoff function, and play the game once.

10. $4.1 $5.1 $3.8 $5.1 $4.6 q A = 64q A = 48 American Airlines q U = 48 q U = 64 United Airlines  The normal-form representation of a game is the payoff matrix (profit matrix).

11.  We can precisely predict the outcome of any game in which every player has a dominant strategy: a strategy that produces a higher payoff than any other strategy the player can use for every possible combination of its rivals’ strategies.  Although firms do not always have dominant strategies, they have them in our airline game.

12. $4.1 $5.1 $3.8 $5.1 $4.6 q A = 64q A = 48 American Airlines q U = 48 q U = 64 United Airlines  A striking feature of this game is that the players choose strategies that do not maximize their joint profit.

13.  In this type of game – called a prisoner’s dilemma game – all players have dominant strategies that lead to a profit that is inferior to what they could achieve if they cooperated and pursued alternative strategies.

14.  In games where not all players have a dominant strategy, we cannot precisely identify the outcome of the game from what we know so far.  Nonetheless, we can determine the outcome of this game by generalizing our earlier logic.  Because we know that a firm will not use a strategy that is strictly dominated by another strategy, we can eliminate any strictly dominated strategy.

15. $0 $3.1 $2.0 $3.1 $4.1 $2.3 $4.6 $3.8 $5.1 $4.6 $2.3 $5.1 $3.8 $4.6 q A = 96q A = 64 American Airlines q A = 48 q U = 64 q U = 96 United Airlines q U = 48

16.  For any given set of strategies chosen by rivals, a player wants to use its best response: the strategy that maximizes a player’s payoff given its beliefs about its rival’s strategies that a rival might use.  A set of strategies is a Nash equilibrium if, when all other players use these strategies, no player can obtain a higher payoff by choosing a different strategy.  If each player uses a Nash equilibrium strategy, then no player wants to deviate by choosing another strategy.

17. $4.1 $5.1 $3.8 $5.1 $4.6 q A = 64q A = 48 American Airlinesq U = 48 q U = 64 United Airlines

18. $0 $1 $0 -$1 Do Not EnterEnter Firm 1 Enter Do Not Enter Firm 2  Each player chooses a single action.

19.  The player chooses among possible actions according to probabilities it assigns.  When both firms enter with a probability of one-half, there is a Nash equilibrium in mixed strategies because neither firms wants to change its strategy, given that the other firm uses its Nash equilibrium mixed strategy.  If both firms use this mixed strategy, each of the four outcomes in the payoff matrix is equally likely.

20.  Thus Firm 1’s expected profit is  Given that Firm 1 uses this mixed strategy, Firm 2 cannot achieve a higher expected profit by using a pure strategy.  If Firm 2 uses the pure strategy of entering, it earns $1 half the time and loses $1 the other half, so its expected profit is $0.  If it stays out with certainty, Firm 2 earns $0 with certainty.

21.  If Firm 2 believes that Firm 1 will use its equilibrium mixed strategy, Firm 2 is indifferent as to which pure strategy it uses.  Suppose to the contrary that one of the actions in the equilibrium mixed strategy had a higher expected payoff than some other action.  Then it would pay to increase the probability that Firm 2 takes the action with the higher expected payoff.

22.  However, if all of the pure strategies that have positive probability in a mixed strategy have the same expected payoff, then the expected payoff of the mixed strategy must also have that expected payoff.  Thus Firm 2 is indifferent as to whether it uses any of these pure strategies or any mixed strategy over these pure strategies.

23.  In our example, why would a firm pick a mixed strategy where its probability of entering is one-half?  In a symmetric game such as this one, we know that both players have the same probability of entering, θ.  Moreover, for Firm 2 to use a mixed strategy, it must be indifferent between entering or not entering if Firm 1 enters with probability θ.  Firm 2’s payoff from entering is

24.  Its payoff from not entering is  Equating these two expected profits, we find that θ = ½.  Thus both firms using a mixed strategy where they enter with a probability of one-half is a Nash equilibrium.

25.  Suppose that Firm 1 enters with probability θ 1.  If Firm 2 enters, it expects to earn  Similarly, if Firm 2 does not enter, then Firm 2 expects to earn

26.  Using these expectations that are conditioned on Firm 1’s entering with probability θ 1, if Firm 2 enters with probability θ 2, Firm 2’s expected payoff is  To maximize Firm 2’s expected payoff with respect to θ 2, the first-order condition is

27.  Thus, Firm 2’s expected profit is maximized if Firm 1 picks the entry strategy with a probability of one-half.  Given the symmetry of the problem, Firm 2’s strategy is the same as Firm 1’s.  Firm 2 enters with a probability of one-half.

28.  One important reason for introducing the concept of a mixed strategy is that some games have no pure-strategy Nash equilibria.  However, every static game with a finite number of players and a finite number of actions has at least one Nash equilibrium, which may involve mixed strategies.

29. $2 $0 $3 $0 $1 Do Not Advertise Advertise Firm 1 Advertise Do Not Advertise Firm 2 Advertising only takes customers from rivals Advertising attracts new customers to the market $2 $3 $4 $3 $5 Do Not Advertise Advertise Firm 1 Advertise Do Not Advertise Firm 2

30.  In dynamic games, players more sequentially or move simultaneously repeatedly over time, so a player has perfect information about other player’s previous moves.  Dynamic games are analyzed in their extensive form, which specifies the number of players, the sequence in which they make their moves, the actions they can take at each move, the information that each player has about player’s previous moves, and the payoff function over all possible strategies.

31.  We illustrate a sequential-move or two-stage game using the Stackelberg airline model, where American Airline chooses its output level before United Airline does.  We assume that both airlines can choose only output levels of 96, 64, and 48 million passengers per quarter.

32.  The normal-form representation of this game does not capture the sequential nature of the firm’s moves.  To demonstrate the role of sequential moves, we use an extensive-form diagram or game tree, which shows the order of the firm’s moves, each firm’s possible actions at the time of its move, and the resulting profits at the end of the game.

33. (4.6, 2.3) (3.1, 2.0) (0, 0) 96 64 48 United Airlines (5.1, 3.8) (4.1, 4.1) (2.0, 3.1) 96 64 48 United Airlines (4.6, 4.6) (3.8, 5.1) (2.3, 4.6) 96 64 48 United Airlines Profits (π A, π u ) Leader’s decisionFollower’s decision 96 64 48 American Airlines (4.1, 4.1) (4.6, 2.3) (3.8, 5.1) (4.6, 2.3) United Airlines