Topic 4: Fun and Games

Home , Backward induction, Nash equilibrium, Solution concept

DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS ECONOMICS 21

Dartmouth College, Department of Economics: Economics 21, Summer‘02‘02‘02

Economics 21, Summer 2002 Andreas Bentz Based Primarily on Shy Chapter 2 and Varian Chapter 27, 28

Review: Choices and Outcomes

Consumer theory: From a given choice set (e.g. budget set), choose the option (e.g. bundle of goods) that you most prefer. Under certainty, the outcome of choice is certain. » Choose the option that has an outcome that maximizes utility. Under uncertainty, the probability distribution over possible outcomes is known.

» Choose the action (associated with a number of outcomes, where each occurs with given probability) that maximizes expected utility.

Choices and Outcomes, cont’d

Producer theory - two extreme cases:

Perfect competition: From a range of possible prices, choose the price that maximizes profit. Under certainty, the outcome of choice is certain:

» p > MC: zero demand, » p < MC: negative profit, » p = MC: zero profit. Under uncertainty, the probability distribution over possible outcomes is known.

» Maximize expected profit (not covered).

Choices and Outcomes, cont’d

Monopolist: From the price-quantity pairs given by the demand curve, choose the one that maximizes profit. Under certainty, the outcome of choice is certain:

» π = p x q(p) - c(q(p)) Under uncertainty, the probability distribution over possible outcomes is known. » Maximize expected profit (not covered).

Choices against Nature

In these cases, choice is the choice of one agent, from a given set of alternatives that give certain (expected) utility (or profit). The agent’s choice is a game against “nature”: The agent chooses an action (associated with a number of outcomes). Then “nature” chooses the outcome that actually occurs: » Under certainty, nature chooses the single outcome for sure. » Under uncertainty, “nature” chooses one of the possible outcomes (with the probability of that outcome). The agent cannot influence nature’s move in this “game”.

Modeling Interaction

In general, in all social interaction, my choice of action influences your choice (because my action influences your payoff [utility, profit], and your action influences mine). Example (duopoly): How I choose my price depends on how I expect you to choose your price, which depends on how you expect me to choose my price, which depends on how I expect you to choose … because our profits depend on how we both choose prices.

We call these social encounters “games”.

Game Theory

Game theory is the study of such “games”: social interactions between rational agents. “All of economics is a branch of game theory” — (Robert Aumann) We have already analyzed some “games”: In monopoly, there is no (real) interaction: » There is only one agent (and nature). In perfect competition, there is no need to model interaction: » The number of agents is so large that the action of one agent has no effect on the other agents’ payoffs. Principal-Agent analyses: » A (non-trivial) example of game-theoretic analysis.

Dartmouth College, Department of Economics: Economics 21, Summer‘02‘02‘02

Game Theory

John von Neumann and Oskar Morgenstern (1944) Theory of Games and Economic Behavior

A Classification of Games

Simultaneous move games (static games): All players make their choices at the same time. Method of analysis: (usually) games in “normal” (or, “strategic”) form.

Sequential move games (dynamic games): Some players make their choices first, then other players observe these choices and then make theirs, etc. Method of analysis: games in “extensive” form.

Dartmouth College, Department of Economics: Economics 21, Summer‘02‘02‘02

Normal Form Games

Simultaneous Move Games in Normal (Strategic) Form

Normal Form Games

Definition: A normal form game (or, strategic form game) is defined by: the set of players in the game; the strategies (actions) that are available to each player;

» each player chooses one of her available strategies; a strategy profile is a list of the strategies chosen by each player; the payoffs for each player, depending on the choice of action of every player; » i.e. each player’s payoff depends on the strategy profile.

Analogy with “parlor” games: e.g. Pong. 11

Example: The “Price War” Game

Duopolists: player 1 (row), player 2 (column)

Player 2: cut price don’t cut cut price (1, 1) (3, 0) Player 1: don’t cut (0, 3) (2, 2)

This “normal form” (or “strategic form”) of the game captures all the information needed.

The “Price War” Game, cont’d

The normal form captures all the information the definition requires: Players: {1, 2} Available strategies: » player 1: {cut price, don’t cut} » player 2: {cut price, don’t cut} Strategy profiles: Payoffs: player 1 - player 2

» (cut price, cut price) 1 1 » (cut price, don’t cut) 3 0 » (don’t cut, cut price) 0 3 » (don’t cut, don’t cut) 2 2

Equilibrium in Games

What is our prediction for the play of a game? Which strategies will agents choose? What is an appropriate definition of equilibrium in games?

What do we want from an equilibrium concept? Existence: The equilibrium concept should yield a prediction for all games. Uniqueness: The equilibrium concept should yield a unique prediction of equilibrium play in all games.

Dominant Strategies

Suggestion 1: If a player has some strategy that gives her a higher payoff than any other strategy she could choose, regardless of what the other players in the game do, she will choose that strategy.

Such a strategy is called a dominant strategy.

Dominant Strategies, cont’d

Equilibrium prediction: If every player has a dominant strategy, every player will choose that dominant strategy.

Definition: An equilibrium in dominant strategies (or dominant strategy equilibrium) is a strategy profile in which every player chooses her dominant strategy.

This is an intuitively appealing and robust equilibrium concept.

The “Price War” Game, cont’d

What is the dominant strategy equilibrium?

Player 2: cut price don’t cut cut price (1, 1) (3, 0) Player 1: don’t cut (0, 3) (2, 2)

The equilibrium strategy profile in dominant strategies is (cut price, cut price). In this equilibrium the payoffs are: (1, 1).

Fun: “Prisoners’ Dilemma” Game

Relabelling players and strategies in the “price war” game, we get the “prisoners’ dilemma” game (political philosophy, politics):

Prisoner 2: confess lie confess (1, 1) (3, 0) Prisoner 1: lie (0, 3) (2, 2)

The “Advertising” Game

Duopolists 1 and 2 decide on advertising expenditure. 2: low med. high low (1, 1) (0, 3) (0, 2) 1: med. (3, 0) (1, 1) (0, 3) high (2, 0) (3, 0) (1, 1)

What is the dominant strategy equilibrium in this game?

The “Advertising” Game, cont’d

In this game, no player has a dominant strategy. There is no dominant strategy equilibrium.

What should our equilibrium prediction be? (Most games do not have a dominant strategy equilibrium.)

Nash Equilibrium

Suggestion 2: If there is a (potential equilibrium) strategy profile in which no player wishes to deviate unilaterally (i.e. choose a different strategy while all other players continue playing their (potential equilibrium) strategies), this will be the equilibrium of the game.

Definition: An equilibrium in which no player wishes to deviate unilaterally is called a Nash equilibrium (John Nash, 1951).

The “Price War” Game, cont’d

What is the Nash equilibrium in the “price war” game? Player 2: cut price don’t cut cut price (1, 1) (3, 0) Player 1: don’t cut (0, 3) (2, 2)

Check each potential equilibrium strategy profile.

Nash E. and Dominant Strategies

Proposition: Every dominant strategy equilibrium is also a Nash equilibrium.

Proof: In a dominant strategy equilibrium, each player is playing the strategy that gives them the highest payoff regardless of what the other players do. Therefore, no player would want to deviate: all other strategies open to the players are worse.

But: not every Nash equilibrium is a dominant strategy equilibrium.

The “Advertising” Game, cont’d

What is the Nash equilibrium in the “advertising” game? 2: low med. high low (1, 1) (0, 3) (0, 2) 1: med. (3, 0) (1, 1) (0, 3) high (2, 0) (3, 0) (1, 1)

The unique Nash equilibrium strategy profile is (high, high).

Existence of Nash Equilibrium

Proposition (Nash): A Nash equilibrium (possibly in mixed strategies) exists in every game.

Mixed strategies are strategies where players “randomize” over strategies. Example (mixed strategy): My advertising expenditure is: low with probability 0.3, medium with prob. 0.2, high with probability 0.5. This course does not cover mixed strategies.

Is the Nash equilibrium prediction unique?

The “Standards” Game

Duopolists decide simultaneously on the standard for VCRs. Sony (2): VHS Beta VHS (2, 1) (0, 0) JVC (1): Beta (0, 0) (1, 2)

What is the Nash equilibrium in this game? There are two Nash equilibria (in pure strat.).

Fun: “Battle of the Sexes” Game

Lovers decide where to go on a Friday night: Her: boxing ballet boxing (2, 1) (0, 0) Him: ballet (0, 0) (1, 2)

Although he prefers boxing, and she prefers ballet, each would rather be with the other than on their own.

Nash Equilibrium and Uniqueness

Nash equilibria are not unique.

Can we somehow trim down the number of Nash equilibria? Thomas Schelling The Strategy of Conflict: » Some equilibria in co-ordination games such as the “battle of the sexes” game are salient. For instance, going wherever he prefers has been salient (is no longer?). The overall conclusion is negative: there is no uncontested way of paring down the number of Nash equilibria. » Are multiple equilibria a feature of the world?

Best Responses

Another way of thinking about Nash equilibria is in terms of “best responses”:

Definition: A player’s best response to the strategies played by the other players, is the strategy that gives her the highest payoff, given the strategies played by the other players.

Best Responses, cont’d

Example: the “price war” game: Player 2: cut price don’t cut cut price (1, 1) (3, 0) Player 1: don’t cut (0, 3) (2, 2)

1’s best response to 2 playing (cut price) is: (cut price).

1’s best response to 2 playing (don’t cut) is: (cut price). 2’s best response to 1 playing (cut price) is: (cut price). 2’s best response to 1 playing (don’t cut) is: (cut price).

Best Responses, cont’d

Example: the “standards” game: Sony (2): VHS Beta VHS (2, 1) (0, 0) JVC (1): Beta (0, 0) (1, 2)

1’s best response to 2 playing (VHS) is: (VHS).

1’s best response to 2 playing (Beta) is: (Beta). 2’s best response to 1 playing (VHS) is: (VHS). 2’s best response to 1 playing (Beta) is: (Beta).

Nash and Best Responses

Proposition: In a Nash equilibrium, every player’s equilibrium strategy is her best response to the other player’s equilibrium strategy.

Proof: In a Nash equilibrium, no player wishes to deviate, given the other players continue to play their Nash equilibrium strategies. Therefore, her strategy must be the best response to the other players’ equilibrium strategies.

Nash and Best Responses, cont’d

Example: the “price war” game: 1’s best response to 2 playing (cut price) is: (cut price). 1’s best response to 2 playing (don’t cut) is: (cut price). 2’s best response to 1 playing (cut price) is: (cut price). 2’s best response to 1 playing (don’t cut) is: (cut price). Player 2: cut price don’t cut cut price (1, 1) (3, 0) Player 1: don’t cut (0, 3) (2, 2)

Nash and Best Responses, cont’d

Example: the “standards” game: 1’s best response to 2 playing (VHS) is: (VHS). 1’s best response to 2 playing (Beta) is: (Beta). 2’s best response to 1 playing (VHS) is: (VHS). 2’s best response to 1 playing (Beta) is: (Beta).

Sony (2): VHS Beta VHS (2, 1) (0, 0) JVC (1): Beta (0, 0) (1, 2)

Dartmouth College, Department of Economics: Economics 21, Summer‘02‘02‘02

Applications

… between monopoly and perfect competition ...

Dartmouth College, Department of Economics: Economics 21, Summer‘02‘02‘02

Market Structure III: An Application

Simultaneous Price Setting: The Bertrand Game (1883) (Shy pp. 107-110)

The Bertrand Game

The game: Players:

» two firms (duopolists), 1 and 2 Strategies:

» players 1 and 2 set prices p1, p2 simultaneously Payoffs:

» players 1, 2 produce quantities y1, y2 of the same homogeneous product, each at constant marginal cost c

» inverse demand: p = a - bY, where Y = y1 + y2

» assumption: if p1 < p2, then y1 = Y, y2 = 0 and vice versa

» assumption: if p1 = p2, then y1 = y2 = 1/2 Y

The Bertrand Game, cont’d

Payoffs, cont’d:

» player i’s profit: πi = piyi -cyi, or πi = (pi -c)yi, where i = 1, 2 » for player 1:

» player 1’s profit when p1 < p2:

• π1 = (p1 -c) Y,

• i.e. π1 = (p1 - c) (a - p1)/b

» player 1’s profit when p1 > p2:

• π1 = 0

» player 1’s profit when p1 = p2:

• π1 = (p1 - c) 1/2 Y,

• i.e. π1 = (p1 - c) 1/2 (a - p1)/b » similarly for player 2.

The Bertrand Game, cont’d

Solution: When prices can be chosen continuously, there is a simple and intuitive solution to the Bertrand game: Can a price less than marginal cost be optimal? » No: profits are negative. Can a price greater than marginal cost be optimal? » Suppose player 1 were to charge a price above marginal cost. Then player 2 could just undercut player 1’s price and take the entire market. Similarly for player 2. The only price at which one player does not have to anticipate being undercut by the other player is price = marginal cost. The Nash equilibrium strategy profile in the Bertrand

game is for both players (i = 1, 2) to set pi = c.

The Bertrand Game, cont’d

If oligopolists compete in prices (“Bertrand competition”), the outcome will be efficient: price = marginal cost.

Dartmouth College, Department of Economics: Economics 21, Summer‘02‘02‘02

Market Structure IV: An Application

Simultaneous Quantity Setting: The Cournot Game (1838) (Shy pp. 98-101; Varian Ch 27)

The Cournot Game

The game: Players:

» two firms (duopolists), 1 and 2 Strategies:

» players 1 and 2 set quantities y1, y2 simultaneously Payoffs:

» players 1, 2 produce quantities y1, y2 of the same homogeneous product, each at constant marginal cost c

» inverse demand: p = a - bY, where Y = y1 + y2

The Cournot Game, cont’d

Payoffs, cont’d:

» firm 1’s profit when it sets quantity y1 and firm 2 sets quantity y2:

• π1 (y1, y2) = p y1 -c y1, or:

• π1 (y1, y2) = (a - b(y1 + y2)) y1 -c y1, or: 2 • π1 (y1, y2) = ay1 -by1 -by2y1 -c y1, or: 2 • π1 (y1, y2) = - by1 + (a - by2 -c)y1. » Similarly for firm 2: 2 • π2 (y1, y2) = - by2 + (a - by1 -c)y2.

The Cournot Game, cont’d

So: 2 Firm 1 profit: π1(y1, y2) = - by1 + (a - by2 -c)y1. 2 Firm 2 profit: π2(y1, y2) = - by2 + (a - by1 -c)y2. What is firm 1’s best response (“reaction”) when firm 2

chooses y2?

Choose y1 to max π1 (y1, y2):

∂π1(y1, y2) / ∂y1 = - 2by1 + a - by2 -c = 0

that is: y1 = (a - by2 -c)/2b This is firm 1’s best response (or “reaction”) function. What is firm 2’s best response function?

y2 = (a - by1 -c)/2b

The Cournot Game, cont’d

Recall: In a Nash equilibrium, every player’s equilibrium strategy is her best response to the other player’s equilibrium strategy. So we know that

y1 = (a - by2 - c)/2b and

y2 = (a - by1 -c)/2b are both true.

Solve for y1:

y1 = (a - c)/3b

y2 = (a - c)/3b

The Cournot Game, cont’d

f1(y2) - firm 1’s best response (or, “reaction”) function

f2(y1) - firm 2’s reaction function

The Cournot Game: Equilibrium?

f1(y2) - firm 1’s best response (or, “reaction”) function

f2(y1) - firm 2’s reaction function

The Cournot Game: Equilibrium!

f1(y2) - firm 1’s best response (or, “reaction”) function

Nash equilibrium

f2(y1) - firm 2’s reaction function

The Cournot Game: Comparison

f1(y2) - firm 1’s best response (or, “reaction”) function Perfect Competition (assuming linear demand and symmetry) Nash equilibrium in the Cournot game

f2(y1) - firm 2’s reaction Monopoly solution function (firm 1 is monopolist)

The Cournot Game, cont’d

If oligopolists compete in quantities (“Cournot competition”), the joint quantity is: greater than the quantity in a monopoly, but less than the quantity under perfect competition (or under Bertrand competition).

Cournot Bertrand quantity Monopoly P. C.

Dartmouth College, Department of Economics: Economics 21, Summer‘02‘02‘02

Extensive Form Games

(Mostly) Sequential Move Games in Extensive Form

Example: The “Entry” Game

potential entrant (1) enter stay out

incumbent (2) (0, 8)

fight share

(-1, -1) (2, 2)

Extensive Form Games

Definition: An extensive form game is: a game tree (one starting node, other decision nodes, terminal nodes, and branches linking each decision node to successor nodes); the set of players in the game; at each decision node, the name of the player making a decision at that node; the actions available to players at each node;

» a player’s strategy is a list of actions of that player at each decision node where that player can take an action; the payoffs for each player at each terminal node.

Extensive Form Games, cont’d

Note: We now need to be careful about the distinction: action - strategy: » An action at some decision node is a player’s decision of what to do when that node is reached. » A strategy is a complete list of actions that a player plans to take at each decision node, whether or not that node is actually reached. » Example (the “entry” game): if player 1 chooses to stay out, player 2’s decision node is not reached.

The “Entry” Game and Nash Eq.

What is the Nash equilibrium in the “entry” game?

Recall: In a Nash equilibrium, no player wishes to deviate unilaterally.

The “Entry” Game, cont’d

potential possible Nash entrant (1) equilibria: enter stay out (enter, fight)

incumbent (2) (0, 8) (enter, share) (stay out, fight) fight share (stay out, share) This game has two Nash (-1, -1) (2, 2) equilibria (in pure strategies).

The “Entry” Game, cont’d

We can convert this extensive form game into a normal (strategic) form game:

potential entrant (1) normal (strategic) form: enter stay out

incumbent (2) (0, 8) fight share

enter (-1, -1) (2, 2) fight share

stay out (0, 8) (0, 8) (-1, -1) (2, 2)

The “Entry” Game, cont’d

One of the two Nash equilibria in the “entry” game is “unreasonable”: (stay out, fight) The potential entrant only stays out because, if she were to enter, the incumbent threatens to fight. But consider what would happen if the entrant did enter: once she has entered (i.e. once we are at player 2’s decision node), the incumbent would want to share the market (i.e. not fight). This Nash equilibrium is based on a “non-credible threat”. This (overall) equilibrium is unreasonable because, once play of the game has reached player 2’s decision node, subsequent play (i.e. play in the subgame that starts at player 2’s decision node) is not an equilibrium.

Multiple Nash Equilibria

In extensive form games we can sometimes eliminate “unreasonable” Nash equilibria. Remember: we want a unique prediction for the play of the game.

We only admit “reasonable” Nash equilibria: We want equilibrium play in a game to be such that each player’s strategies are an equilibrium not only in the overall game, but also at every decision node, for the subsequent game (the subgame starting at that decision node).

Subgame Perfect Equilibrium

Definition: A subgame is the game that starts at one of the decision nodes of the original game; i.e. it is a decision node from the original game along with the decision nodes and terminal nodes directly following this node.

Definition: A Nash equilibrium with the property that it induces equilibrium play at every subgame is called a subgame perfect equilibrium.

The “Entry” Game, cont’d

potential 1. What is the entrant (1) equilibrium in the enter stay out subgame starting at player 2’s incumbent (2) (0, 8) decision node? 2. Once we know this, what is the fight share equilibrium in the subgame starting (-1, -1) (2, 2) at the starting node?

The “Entry” Game, cont’d

There is a unique subgame perfect equilibrium in the “entry” game.

Subgame perfection may help us trim down the number of Nash equilibria in sequential- move games in extensive form.

Subgame perfection is the solution concept we will use for sequential-move games in extensive form.

Backward Induction

A method for finding subgame perfect equilibria is backward induction. A subgame perfect equilibrium is a specification of all players’ strategies such that play in every subgame is a (Nash) equilibrium for that subgame. In particular, this is true for the final subgame(s). So we know what happens in the final subgame: we can replace that subgame by the payoff that will be reached in that subgame. Then proceed similarly in this new “reduced” game, until there is only one subgame left.

Backward Induction, cont’d

potential entrant (1) enter stay out

incumbent (2) (0, 8)

fight share

(-1, -1) (2, 2)

Dartmouth College, Department of Economics: Economics 21, Summer‘02‘02‘02

Market Structure V: An Application

Entry Deterrence Dixit (1982) AER

Entry Deterrence

Entry deterrence: the incumbent takes an action that influences payoffs such that she can “commit” to the threat of fighting a new entrant. Remember: in the “entry” game, the threat to fight was non- credible, and was therefore eliminated by subgame perfection. Suppose before playing the “entry” game, the incumbent can choose to incur a cost in readiness to fight a price war. Suppose this cost does not reduce payoffs if there is a price war, but does reduce costs if there is no price war. (In our example, this cost is 4.) What is the subgame perfect equilibrium? Solve by backward induction. 66

The “Entry Deterrence” Game

incumbent (2) committed passive

potential potential entrant (1) entrant (1) enter stay out enter stay out

incumbent (2) (0, 4) incumbent (2) (0, 8) fight share fight share

(-1, -1) (2, -2) (-1, -1) (2, 2)

“Entry Deterrence” Game, cont’d

The “entry deterrence” game in our example has a unique subgame perfect equilibrium: (stay out [at B], enter [at C]; committed [at A], fight [at D], share [at E]). (Remember: a player’s strategy lists an action for each of that player’s decision nodes.)

“Entry Deterrence” Game, cont’d

We can convert this game into a normal (strategic) form game: A incumbent (2) committed passive

BCplayer 1: etc. potential potential entrant (1) entrant (1)

enter stay out enter stay out enter (B), enter (C) DE enter (B), stay out (C) incumbent (2) (0, 4) incumbent (2) (0, 8)

fight share fight share stay out (B), enter (C) stay out (B), stay out (C) (-1, -1) (2, -2) (-1, -1) (2, 2)

Dartmouth College, Department of Economics: Economics 21, Summer‘02‘02‘02

Market Structure VI: An Application

Sequential Quantity Setting: The Stackelberg Game

The Stackelberg Game

The game: Players:

» two firms (duopolists), 1 and 2 Strategies:

» players 1 and 2 set quantities y1, y2 » player 1 moves first (she is the Stackelberg leader)

» player 2 observes 1’s choice of y1, and then sets y2. Payoffs:

» players 1, 2 produce quantities y1, y2 of the same homogeneous product, each at constant marginal cost c

» inverse demand: p = a - bY, where Y = y1 + y2

The Stackelberg Game, cont’d

Payoffs, cont’d:

» firm 1’s profit when it sets quantity y1 and firm 2 sets quantity y2:

• π1 (y1, y2) = p y1 -c y1, or:

• π1 (y1, y2) = (a - b(y1 + y2)) y1 -c y1, or: 2 • π1 (y1, y2) = ay1 -by1 -by2y1 -c y1, or: 2 • π1 (y1, y2) = - by1 + (a - by2 -c)y1.

» The combinations of y1 and y2 for which profit is constant are firm 1’s isoprofit curves. (Topic 4) » Similarly for firm 2: 2 • π2 (y1, y2) = - by2 + (a - by1 -c)y2.

The Stackelberg Game, cont’d

Solution: by backward induction: Player 2 chooses the quantity that is best for her, after observing what player 1 has chosen, i.e. player 2 plays her best response to player 1’s choice: player 2 chooses a point on her best response function. Knowing this, player 1 chooses the quantity that is best for her, given that (after she has chosen), player 2 will choose a point on her best response function, i.e. player 1 chooses the point on player 2’s best response function that is best for her. 73

The Stackelberg Game, cont’d

Firm 2 chooses the quantity that is best, after having

observed firm 1’s choice of quantity y1.

Firm 2 chooses y2 to: 2 max π2 (y1, y2) = - by2 + (a - by1 -c)y2.

-2by2 + a - by1 -c = 0, or

y2 = (a - by1 - c)/2b. Knowing this, firm 1 chooses the quantity that is best.

Firm 1 chooses y1 to:

max π1 (y1, (a - by1 - c)/2b) = 2 = - by1 + (a -b((a -by1 - c)/2b) - c)y1.

-2by1 + a - c - 0.5a + 0.5c + by1 = 0, or

y1 = (a - c) / 2b

The Stackelberg Game, cont’d

So firm 1 chooses y1 = (a - c) / 2b.

Therefore firm 2 chooses y2 = (a - by1 - c)/2b, or y2 = (a - b((a - c) / 2b) - c)/2b, or:

y2 = (a - c) / 4b

The Stackelberg Game, cont’d

f1(y2) - firm 1’s best response (or, “reaction”) function

Nash equilibrium in the Cournot game Subgame perfect equilibrium in the Stackelberg game

f2(y1) - firm 2’s reaction function