Economics 335 March 2, 1999

Notes 6: Game TheoryI. Introduction

A. Idea of Game Theory

Game theory analyzes interactions between rational, decision-making individuals who may not be able topredict fully the outcomes of their actions.

B. Definition

A game is a formal representation of a situation in which a number of decision makers (players) interact in asetting of strategic interdependence. By that, we mean that the welfare of each decision maker depends notonly on her own actions, but also on the actions of the other players. Moreover, the actions that are best for herto take may depend on what she expects the other players to do.

C. Elements of a Game

1. Players or decision makers

a. firmsb. consumersc. poker or chess playersd. nature - nature chooses actions according to fixed probabilities

2. Rules of the game

a. Who moves when?b. What do players know when they move (information structure)?c. What options are available to players at various points of the game?

3. Outcomes

For each possible set of actions (strategies) by the players, what is the outcome of the game?

4. The payoffs

What are the player’s preferences over the outcomes? How do they rank these outcomes?

D. Example 1 - Tick-Tack-Toe

1. Players

There are two players denoted X and O.

2. Rules of the game

a. The players are faced with a board that consists of nine squares arrayed with three rows ofthree squares stacked on each other as in Figure 1.

b. The players take turns putting their symbol (X or 0) into an as yet unmarked square.

c. Player X moves first.

d. Both players observe all choices previously made.

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3. Outcomes

a. The first player to have three marks in a row (horizontally, vertically, or diagonally) wins andreceives $1.00 dollar from the other player.

b. If no one succeeds in winning after all nine boxes are marked, the game is a tie and nopayments are made or received by either player.

4. Payoffs

The payoff is the amount of money received and we assume that the players prefer more money toless money.

Figure 1

E. Example 2 - Turkey sales

1. Players

There are two players denoted Hy-Vee and Fareway

2. Rules of the game

a. The game is played 7 days before the Thanksgiving Holiday.

b. Both stores must deliver a weekly advertising supplement to the local paper stating the priceper pound of frozen whole turkeys.

c. The turkeys advertised are of the same quality.

d. The players may only announce one of two prices : low and high.

e. The stores do not know which price will be submitted by the other store, and the deadline forgetting the advertisement in the paper is the same for both stores.

3. Outcomes

The outcomes are specified using a matrix listing the net returns for each possible combination of choicesby each firm where the first number in each cell represents the net returns to Fareway.

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Table 1 Outcomes for Fareway and Hy-Vee from AdvertisingDecision

Hy-Vee’s Strategy

Low Price High Price

Low Price 1000 1000 1500 500

Fareway’s Strategy

High Price 500 1500 1200 1200

The idea is that if a store advertizes a high price and the other store advertizes a low price, the low pricestore will steal customers from the high price store.

4. Payoffs

The payoff is the amount of money received and we assume that the players prefer more money to lessmoney.

II. Strategic or “Normal” Form Games

A. Timing

In a normal form game all players are assumed to make their moves at the same time. The pricing game aboveis in normal form while tick-tack-toe is not.

B. Formal Description of the Game

1. A set of N players whose names are listed in the set I = {1,2,..., N). For the turkey example the set Iis I = {1, 2} where 1 is Fareway and 2 is Hy-Vee.

2. For each player i, i I, there is a strategy set Si which contains all actions (or pure strategies)available to player i. We sometimes denote this set as the action set Ai. A particular element of thisset is denoted si. At this point we assume that the strategy set is finite. The strategy set for eachplayer in the turkey pricing game is Si = {low, high}.

3. An outcome of the game is a listing of the strategies chosen by each player and is denoted by s = (s1,s2, ...,si, ... sN-1, sN). The possible outcomes in the turkey pricing game are (low, low), (low, high),(high, low) and (high, high). We denote the set of all actions except that of ith player as s¬i = (s1, s2,...,si-1, si+1, ... sN-1, sN).

4. Each player has a payoff function ui (s1, s2, ...,si, ... sN-1, sN) which assigns a real number to eachoutcome of the game. Formally, each payoff function ui maps the N-dimensional vector, s, into thereal line. It is assumed that players prefer higher numbers to lower ones. The payoff functions inthe turkey pricing game are as follows:

u1 ( low,low) = 1000 u2(low,low) = 1000u1 ( low, high) = 1500 u2(low,high) = 500u1 ( high,low) = 500 u2(high,low) = 1000u1 ( high, high) = 1200 u2(high,high) = 1200

It is clear in this game that total net returns are highest if both firms choose high prices while totalreturns are the same for all other choices.

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u i (s̃ i,s ¬i) > u i (s i,s ¬i), for every s i g S i

C. Equilibrium of the Game using Dominant Strategies

1. Definition of dominant strategies

A particular strategy g Si is said to be a dominant strategy for player i if no matter what all others̃ i

players choose, playing always maximizes player i’s payoff. Formally, for every choice of alls̃ i

strategies for all players except i, s¬i,

We say that a strategy is a dominated strategy for a player if there exists another strategy which isalways strictly better, no matter what strategies are chosen by the other players.

A weakly dominant strategy for a player in one that is weakly better (as good as or better) than everyother strategy for that player, no matter what strategies are chosen by the other players.

A strategy is weakly dominated for a player if there exists another strategy which is always weaklybetter, no matter what strategies are chosen by the other players.

2. Example

Consider Fareway in the turkey pricing game. If Hy-Vee chooses a low price then Fareway will get 1000if it chooses a low price and 500 if it chooses a high price so Fareway will choose a low price if it knowsHy-Vee has chosen a low price. If Hy-Vee chooses a high price, Fareway will get 1500 if it chooses alow price and 1200 if it chooses a high price so Fareway will choose a low price if it knows Hy-Vee willchoose a high price. Therefore the optimal strategy for Fareway in either case is a low price. Thereforewe say that a low price is a dominant strategy for Fareway.

If we were to perform the same exercise for Hy-Vee we would find that Hy-Vee also has a dominantstrategy of choosing the low price.

3. Definition of equilibrium in dominant strategies

An outcome is said to be an(s̃ 1, s̃ 2, ... ,s̃ N ) where s̃ i g S i for every i ' 1,2, ...,N

equilibrium in dominant strategies if is a dominant strategy for each player i.s̃ i

4. Example of dominant strategy equilibrium

In the above turkey pricing game, the outcome (low, low) is a dominant strategy equilibrium since “low”is a dominant strategy for both Hy-Vee and Fareway. This result is somewhat surprising since if bothstores were to advertise the high price, they would be better off. The point (high, high) is called Paretosuperior to the point (low, low) since both firms are better off at (high, high).

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5. Situations with no dominant strategy

Consider a different pricing game, this time between two local agricultural cooperatives who are placingadvertisements for tires in the local newspaper. The tires are the same brand (Unmarked and BlandIncorporated) and style. Let the two firms be denoted FCS (Farm Coop Society) and LCI (Local CoopIncorporated). Assume that there are 3 elements of the strategy set (low, medium, high) for eachcooperative. The outcome matrix for this problem is as follows.

Table 2 Outcomes for FCS and LCI from Advertising Decision

LCI’s Strategy

Low Price Medium Price High Price

Low Price 1000 1000 1600 1200 2000 1500

FCS’s Strategy Medium Price 1200 1600 2000 2000 2500 1800

High Price 1500 2000 1800 2500 2200 2200

The idea is that a store’s returns will go up as the other store raises its price. However, holding the otherstore’s price constant, a store’s revenue may go up as it raises its own price if sales do not fall too much. For example when LCI’s price is low, FCS does better by raising its price from low to medium and frommedium to high. When LCI’s price is medium, FCS does better by raising its price from low to medium,but does worse by raising it from medium to high, since sales fall more than is compensated for by thehigher price.

Consider now the optimal strategies for FCS. If LCI chooses a low price, FCS prefers a high price. IfLCI chooses medium, FCS does best with a medium price, while if LCI chooses high, FCS also does bestwith a medium price strategy. Thus there is no dominant strategy for FCS since sometimes it prefers amedium price and sometimes it prefers a high price. We can say, however, that the low price strategy isdominated since it is never optimal. If we perform the same exercise for LCI, we find that when FCSchooses a low price , LCI does best with a high price, while when FCS chooses either medium or high,LCI does best with a medium price.

6. Iterated dominant strategy equilibrium

In the tire pricing game, there is no dominant strategy equilibrium since at least one player does not havea dominant strategy. But this game still has an equilibrium that can be found by iterating the decisionprocess.

Since FCS knows that it will never choose a low price, we can eliminate the first row of Table 2. Andsince LCI will never choose a low price, we can also eliminate the first column. The information isreproduced in Table 3.

Table 3 Outcomes for FCS and LCI from Advertising Decision with LowPrices Excluded

ICI’s Strategy

Medium Price High Price

FCS’s Strategy Medium Price 2000 2000 2500 1800

High Price 1800 2500 2200 2200

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u i (s̃ i, s̃ ¬i) $ u i (s i, s̃ ¬i), for every s i g S i

Now consider the high price strategy for FCS. It is dominated by the medium price strategywhether LCI chooses medium or high so the medium price strategy is dominant for FCS. We canalso show that the medium price strategy is dominant for LCI. Thus the equilibrium of this gameand the one in Table 2 is (medium, medium).

D. Equilibrium of a Game Using the Concept of Nash Equilibrium

1. Games with no equilibrium in dominant strategies

Many games have no equilibrium in dominant strategies, particularly if one player does not have adominant strategy. In many cases, there is not an iterated equilibrium either.

2. Example - Standards for power takeoff (PTO) connections

Consider a game played by John Deere and New Holland. Assume that John Deere is considering a newstandard for power takeoff connections for its next model year tractors. We will label the new standard asnew and the current one as old in the following matrix representation of the game. While not totallyrealistic we will assume that this is a game of simultaneous moves in that the dealers must come to marketbefore knowing which standard the other has adopted.

Table 4 Outcomes for John Deere and New Holland Adoption Decision on PTO standards

John Deere’s Strategy

Old Standard New Standard

New Holland’s Strategy Old Standard 200 1000 100 800

New Standard 100 800 300 1200

First consider if there is an equilibrium in dominant strategies. Since New Holland prefers old whenJohn Deere chooses old and new when John Deere chooses new, there is no equilibrium in dominantstrategies. Since there are only two choices, the problem has no meaning if we eliminate one of thestrategies. Thus there is no iterated equilibrium in dominant strategies . Given what we know sofar, we cannot predict an equilibrium for this game.

3. Definition of Nash equilibrium

The most common equilibrium concept used in game theory is due to John Nash and is called Nashequilibrium. The idea is that an outcome is an equilibrium if no player would find it to heradvantage to deviate from this point. Specifically:

Suppose that Then the strategy profiles̃ i g S i for every i ' 1,2, ...,N.

is said to be a Nash equilibrium (NE) if no player would benefit bys̃ ' (s̃ 1, s̃ 2, ... ,s̃ N )deviating, provided that no other player deviates either. Formally, for every player 1, i = 1, 2,... , N,

Notice the difference between this and a dominant strategy equilibrium is that the second argument

in . The strategy need not be better in all circumstancesu i (@ , @) is s̃¬i and not s ¬i s̃ i

(against all possible strategies ), but merely better against the specific strategy profile ,s ¬i s̃ ¬i

which is played in the equilibrium.

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4. Example

Now investigate the PTO game and Nash equilibrium. Consider the outcome (old, old). If John Deerechooses old then the best strategy for New Holland is to choose old. If New Holland chooses old, thebest strategy for John Deere is to choose old. Thus the outcome (old, old) is a Nash equilibrium sinceneither player has an incentive to deviate from it. The same is true of the (new, new) strategy. Thestrategy (old, new) is not a Nash equilibrium since if John Deere chooses new, New Holland will want tochoose old. Similarly for the outcome (new, old). Thus this game has 2 Nash equilibriums and wecannot really predict which will be chosen.

5. Nash equilibrium and dominant strategy equilibrium

It can be easily shown that any dominant strategy equilibrium is a Nash equilibrium, but not vice versa.

E. Non-existence of Nash Equilibrium

1. Definition

Many games have no equilibrium in dominant or iterated dominant strategies and may not even have aNash equilibrium in pure strategies. (Players do not randomize their choice of actions as, for example, abaseball pitcher does when choosing which sort of ball to throw.) A game has no equilibrium in purestrategies if at every possible outcome, at least one player has an incentive to deviate.

2. Example

Consider the following game between Farmer Slack and Freddy Foreclose. The game involves the twoplayers deciding where to spend the morning. They both leave their homes at the same time. Freddy’spayoffs are higher if he ends up at the same place as Farmer Slack since he can then deliver theforeclosure papers. Farmer Slack is better off if she can avoid Freddy. Both of them prefer the CoffeeShop to the Corn Silage Pit. Consider then the matrix of payoffs for this (cat and mouse) game.

Table 5 Outcomes for Farmer Slack and Freddy Foreclose from thedecision of where to spend the morning.

Farmer Slack

Silage Pit Coffee Shop

Freddy Foreclose Silage Pit 400 0 0 500

Coffee Shop 100 200 1000 50

There are four possible outcomes of this game. Consider each of them in turn.

a. (Silage Pit, Silage Pit) For this outcome uSlack (Silage Pit, Coffee Shop) = 500 > 0 = uSlack

(Silage Pit, Silage Pit) so if Freddy plays silage pit, Slack will play Coffee Shop. Thus this isnot a Nash equilibrium.

b. (Silage Pit, Coffee Shop) For this outcome uFreddy (Coffee Shop, Coffee Shop) = 1000 > 0 = uFreddy (Silage Pit, Coffee Shop) so if Slack plays coffee shop, Freddy will play Coffee Shop. Thus (Silage Pit, Coffee Shop) is not an equilibrium.

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400p ' 100p % 1000(1&p) Y p '1013

E( Freddy|p'.60) ' .6q (400)% .6 (1&q) (100)% .4q (0)% .4 (1&q) (1000)' 240q%60&60q% 0%400&400q' 460&220q

c. (Coffee Shop, Silage Pitt) For this outcome uFreddy (Silage Pit, Silage Pit) = 400 > 100 = uFreddy (Coffee Shop, Silage Pit) so if Slack plays Silage Pit, Freddy will play Silage Pit. Thusthis is not an equilibrium.

d. (Coffee Shop, Coffee Shop) For this outcome uSlack (Coffee Shop, Silage Pit) = 200 > 50 = uFreddy (Silage Pit, Coffee Shop) so if Freddy plays coffee shop, Slack will Silage Pit. Thus(Coffee Shop, Coffee Shop) is not an equilibrium.

This game then has no Nash equilibrium outcome.

F. Mixed Strategy Equilibrium

Although this game does not have an equilibrium in pure strategies, if we expand our concept of a strategy, itdoes have an equilibrium. The reason why we could find no equilibrium before is that, whatever Farmer Slackdoes, Freddy Foreclose will want to be in the same place as Slack, but then Slack will want to move to the otherplace. As long as Freddy knows where Slack is, Slack gets a bad payoff.

Therefore, what Slack really must do, in order to do well, is to be unpredictable. She must randomize herchoice to keep Freddy guessing. We call such a strategy a mixed strategy. Freddy will also want to play amixed strategy. As long as Slack knows where Freddy is, she will choose to go elsewhere. To have any chanceof finding Slack, Freddy must make sure that Slack does not know exactly where he will be. In general, as inthis game, when one player has a mixed strategy in the equilibrium, so will the other.

To have mixed strategy equilibrium, it must be the case that the players are indifferent between the actionswhich they choose to play. If a player does strictly better with one action or another, then he will not want toplay a mixed strategy. Thus we solve for the mixed strategy equilibrium by finding the randomization choicewhich makes the other player indifferent between actions.

In this case, suppose that Farmer Slack chooses the silage pit with probability p, and chooses the coffee shopwith probability 1-p. Then by choosing the silage pit, Freddy gets the payoff 400p + 0(1-p) = 400p, and bychoosing the coffee shop, Freddy gets 100p + 1000(1-p). For Freddy to be indifferent between the silage pitand the coffee shop given Farmer Slack’s choice of p, it must be that p satisfies

Thus, to keep Freddy guessing, Slack must go to the silage pit 10/13 of the time, and go to the coffee shop 3/13of the time. If Farmer Slack goes to either place more often than this, then Freddy will prefer one action overthe other, and it will not be an equilibrium. (Since if Freddy does one action all the time, Slack will not want torandomize.)

We can see this another way by considering the best strategy for Farmer Slack given her belief’s about whatFreddy thinks she will do. Suppose Farmer Slack thinks that Freddy thinks that she will choose the silage pitwith probability .60 Then she can compute Freddy’s expected payoffs as follows (assuming he chooses thesilage pit with probability q).

Freddy will maximize his payoffs by setting q equal to 0 and have expected payoff equal to 460. Thus eventhough Freddy knows Slack will be in the Silage Pit 60% of the time, it is such an undesirable place he would

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E( Freddy|p'.80) ' .8q (400)% .8 (1&q) (100)% .2q (0)% .2 (1&q) (1000)' 320q%80&80q% 0%200&200q' 280%40q

E( Freddy|p'1013

) '1013

q (400)%1013

(1&q) (100)%313

.2q (0)%313

(1&q) (1000)

'400013

q% 100013

&100013

q% 0% 300013

&300013

q

'400013

' 307.692

200(1&q) ' 500q % 500(1&q) Y q '3

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rather just stay at the coffee shop and have an expected payoff of 460 (400 from the 40% of the time Slackcomes and 60 from the 60% of the time she doesn’t). This is not a random strategy and so her expectation thatFreddy will play a random strategy is wrong. Given that Slack has figured out that Freddy will just stay in thecoffee shop if she plays silage pit with probability .6, she will now not play a random strategy but play silagepit all the time. But now Freddy will play silage pit and we are back to where we started.

Now suppose Slack thinks that Freddy thinks that she will randomize with probability of .80 on the silage pit. Freddy’s expected profits are now as follows.

Freddy now does best by making q as large as possible or choosing the silage pit with probability 1. This givesan expected payoff of 320 which is less than last time but more than if he chose the coffee shop with anypositive probability. Even though Freddy hates the silage pit he will stay there all the time for the 80% chanceof getting Slack. Again Freddy does not randomize but now chooses the silage pit with certainty. Only whenp = 10/13 will Freddy not choose q equal to zero or one. Specifically

In this case any value of q will give the same expected payoff and so Freddy can randomize with no change inthe expected return.

We can also calculate what Freddy’s strategy must be. Suppose that Freddy goes to silage pit with probabilityq. Then, by going to the silage pit, Slack gets a payoff of (0)q + 200(1-q), and by going to the coffee shop, shegets 500q + 50(1-q). For Slack to be indifferent or to sometimes do one and sometimes do the other, we musthave

In the equilibrium, Freddy spends 3/13 of the time at the silage pit, and spends 10/13 of the time at the coffeeshop.

What is the probability that Farmer Slack will be caught by Freddy Foreclose?

Example: Another real-world example of a mixed strategy equilibrium is in the game of baseball. The pitcheralways wants to keep the batter guessing about what kind of ball she will throw. If the batter knows which typeof ball to expect, he can tailor his response to this kind of ball and hit a home-run. By randomizing, the pitcherforces the batter to choose some sort of generic response, or forces the batter to randomize between differenttypes of responses, in the hope that at least some of the time his response will be appropriate and he will belucky enough to hit a home run anyway.

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G. The Path to Nash Equilibrium

In the games we have to this point we assume that there is perfect information in that all the players know thepayoffs of all other players. Given this fact we can sometimes predict the equilibrium of the game byconsidering alternative hypothetical beliefs for each player and checking whether they lead to a Nashequilibrium.

Consider the following game between two competing feed companies who are planning to run a sale on aparticular premix. Assume that the two companies are Clark’s Feeds and Fenman’s Feeds. The payoffs toalternative strategies are given in Table 6 where the first element of a box is the payoff to Clark. Clark has areputation of sometimes knocking a little off the sale price (with sufficient negotiating and small talk) whileFenman has never been known to give a price break to anyone.

Given this fact, Clark will always do better by advertising the same price as Fenman, since some customers willcome because they know they can beat the price advertised little bit by negotiating. On the other hand, Fenmanoften does best by not advertising the same price as Clark. For example, Fenman does best by advertising lowwhen Clark advertises low, and does best by advertising medium when Clark advertises high. However, itturns out to be better to advertise high when Clark advertises medium since some customers don’t like hagglingover small price breaks, and besides, Clark has a reputation for occasionally watering the feed. Consider thetable that follows.

Table 6 Outcomes for Clark and Fenman from Advertising Decision

Fenman’s Strategy

Low Price Medium Price High Price

Low Price 100 200 150 150 200 100

Clark’s Strategy Medium Price 80 220 200 225 250 230

High Price 75 250 150 260 280 250

There is no equilibrium in dominant strategies since when Clark picks low, Fenman will pick low, but whenClark picks medium, Fenman will pick high. Thus Fenman has no dominant strategy. The same is true forClark. You can verify that there is no iterated dominant strategy equilibrium either. Now consider how to finda Nash equilibrium sequentially. Because Clark is rational, he will choose the strategy that is best, given somebelief about the strategy that Fenman will choose. And because Clark believes that Fenman is also rational, heknows that Fenman will choose the best strategy given some belief about what Clark will do.

So start by supposing (for the sake of argument) that Clark believes that Fenman thinks that Clark willadvertise a high price. In this case, Clark believes that Fenman will advertise a medium price, since 260 is thelargest payoff for Fenman when Clark advertises a high price. But if Fenman advertise a medium price, thenClark does best by advertising a medium price, not a high price. Thus Fenmans’ hypothesis turns out to bewrong. Clarks’ conjecture about what Fenman believes about him is inconsistent with optimal behaviorfor each player and thus is not rational.

The result is similar if Clark believes that Fenman thinks that Clark will advertise a medium price. If Fenmanthinks Clark will advertise a medium price, then Fenman will advertise a high price. But if Fenman advertisesa high price, Clark does best by advertising a high price, not a medium price, since 280 is the largest payoff forClark when Fenman chooses a high price. Thus Clark’s conjecture is not consistent with rational behavioron the part of each player.

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But now suppose that Clark believes that Fenman thinks that Clark will advertise a low price. In this case ifFenman thinks Clark will advertise a low price, then Fenman will advertise a low price as well. And if Fenmanadvertises a low price, Clark does best by advertising a low price, since 100 is the largest payoff for Clarkwhen Fenman advertises a low price. Thus Clark does exactly what Fenman expects Clark to do. Clark’sconjecture is consistent with rational behavior on the part of each player. We have an equilibrium.

We can summarize this process for Clark in a table:

Table 7 Clark’s best strategy for each belief he could have about Fenman

Clark’s belief about Fenman’sbelief about his strategy

Clark’s belief about Fenman’soptimal strategy

Clark’s best strategy

Low Price Low Price Low Price

Medium Price High Price High Price

High Price Medium Price Medium Price

We can construct a similar table for Fenman summarizing her best strategy considering what she believesClark thinks that she will do.

Table 8 Fenman’s best strategy for each belief she could have about Clark

Fenman’s belief about Clark’sbelief about her strategy

Fenman’s belief about Clark’soptimal strategy

Fenman’s best strategy

Low Price Low Price Low Price

Medium Price Medium Price High Price

High Price High Price Medium Price

The point is that the set of beliefs and actions (low, low) are mutually consistent for both players. This is aNash equilibrium, as you can verify.

H. Best Response Functions

1. Definition

A best response function for a player in an N-person game gives the optimal strategy for a given playergiven the strategies of other players. Specifically, the best response function for player i is the function Ri

(s¬i), which for given strategies s¬i of players 1,2,..., i-1, i+1, ... ,N, assigns a strategy si = Ri (s¬i) thatmaximizes player i’s payoff or utility ui(si, s¬i).

2. Proposition concerning Nash equilibrium

If is a Nash equilibrium outcome, then for every player i.s̃ s̃ i ' R i(s̃¬i)

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R clarks s Fenmans '

low if s Fenmans ' lowmedium if s Fenmans ' medium

high if s Fenmans ' high

R Fenamns s clarks '

low if s Clarks ' lowmedium if s Clarks ' highhigh if s Clarks ' medium

3. Example for the feed pricing game

The best response function for Clarks is as follows:

The best response function for Fenmans is as follows:

We can find a Nash equilibrium, if it exists, by finding points that are on the best response functions ofboth players. In this game there are nine possible outcomes. Consider the outcome (low, medium). ForClarks, RClarks (medium) = medium which is not low so this is not an equilibrium. Now consider theoutcome (high, medium). For Fenmans, RFenmans (high) = medium so this is a best response for Fenmans. But consider whether this is a best response for Clarks. RClarks (medium) = medium which is not high, sothis is not an equilibrium. But consider (low, low). Here RClarks (low) = low and RFenmans (low) = low andthis point is a Nash equilibrium.

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III. Extensive Form Games

A. Idea

Extensive form games are ones in which players make moves in a sequence and may move more than once.Some are finite and some infinite in length. Normal form games played several times (repeated) are a type ofextensive form game.

B. Review of the Elements of a Game

1. Players or decision makers

a. decision-making entitiesb. nature - nature chooses actions according to fixed probabilities

2. Rules of the game

a. Who moves when or the order of play?

1) simultaneous move games2) sequential move games

b. What options are available to players at various points of the game?

1) Decision nodes are points of the game where an individual player is called upon to makea decision.

2) The location of the nodes and the options available at the node are both important

c. What do players know when they move (information structure)?

1) A game has perfect recall if no player forgets any information she once knew, and eachplayers know the actions he or she has taken.

2) If every player at every decision node knows the actions taken previously by every otherplayer (including nature), then the game is one of perfect information. Because gamesof imperfect information are much more difficult to solve, we will study only games ofperfect information here.

3) We also assume that all players are rational and this fact is known by all players.

3. Representation of rules by a game tree

a. Nodes are decision points.

b. Branches represent choices that are open to the decision maker.

c. Every branch connects two nodes, one of which is a parent and one of which is a child.

d. Every node has at most one parent and is connected to the parent by one branch.

e. If a node B comes after node A then node A is called an ancestor of B and B is called asuccessor of A.

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Incumbent

Technology A

Technology B

Entrant

Entrant

Enter

Do not enter

Enter

Do not enter

(1000, -100)

(2000, 0)

(500, 500)

(3000, 0)

Extensive Form Game TreeEntry Deterrence I

I1

E1

E2

T1

T2

T4

T3

f. A node with no ancestors is called an initial node.

g. A node with no successors is called a terminal node. All other nodes (including initial nodes)are called decision nodes.

h. Every game tree has one initial (starting) node. All other nodes are descendants of this one.

i. example game diagram

Consider a game with two players, an incumbent in the industry and a potentialentrant. The incumbent moves first and chooses a technology. The potential entrant thendecides whether to enter or not. Based on the technology chosen by the incumbent and theentry decision, the firms receive different pay-offs.

The initial node is I1. The terminal nodes are T1, T2, T3, T4. Node E1 is a parent andancestor of T2. Node I1 is a parent of E2 and an ancestor of T4. The branches emanatingfrom I1 are Technology A and Technology B while the branches emanating from E1 are“Enter” and “Do not enter”.

4. Outcomes

For each possible set of actions (strategies) by the players, what is the outcome of the game?

The outcomes in the diagram above are represented by the terminal nodes (T1-T4) which are determinedby the strategies chosen by the players.

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5. The payoffs

What are the player’s preferences over the outcomes? How do they rank these outcomes?

The payoffs in the above game are given by the numbers next to the terminal nodes. The first number ineach pair is the payoff to the incumbent at that terminal node. Players prefer more to less.

C. Formal Definition of an Extensive Form Game

1. A game tree containing a starting node, other decision nodes, terminal nodes, and branches linkingeach decision node to a successor node.

2. A list of N $ 1 players, indexed by i, i = 1, 2, 3, ..., N.

3. For each decision node, the name of the player entitled to choose an action.

4. For each player i, a specification of i’s strategy (action ) set at each node that player i is entitled tochoose an action.

5. A specification of the payoff to each player at each terminal node.

D. Strategies for Extensive Form Games

1. Definition

In a normal form game, a strategy and an action are synonymous since actions are simultaneous andplayed only once. In an extensive form game, a player must consider what actions she would take underdifferent circumstances, after different choices of the other players who move earlier in the game.

A strategy for player i (denoted si) is a complete plan (list) of actions, one action for each decision nodeat which the player is entitled to choose an action, corresponding to a set of decisions by other playerswho move earlier.

2. Example for the entry game

Consider the strategies available to the entrant. Since the entrant may end up at either E1 or E2, astrategy is a specification of the precise actions to be taken at both nodes. While it is clear that theentrant will end up at either E1 or E2 and not both, a strategy must specify an action at each of the twonodes. Therefore the entrant has 4 possible strategies in this case. They are denoted (Enter, Enter),(Enter, Don’t Enter), (Don’t Enter, Enter), (Don’t Enter, Don’t Enter) where the first element of the pairdenotes the action at node E1. For example, the first strategy says to enter regardless of the technologychosen by the incumbent. The strategy set for the incumbent consists of the action set at node I1 or (A,B).

3. Writing an extensive form game in normal (strategic) form

Once we specify the complete strategy set for all players, we can represent an extensive form game innormal form by assuming the all players must announce their strategies at the same time. In the entrygame, recall that I has two strategies and E has four strategies. We can express this game in a table, justlike any normal form game:

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Table 8 Outcomes for Entry Game I in Normal Form

Entrant’s Strategy

(E, E) (E, D) (D, E) (D, D)

Technology A (1000, -100) (1000, -100) (2000, 0) (2000, 0)

Incumbent’s Strategy

Technology B (500, 500) (3000, 0) (500, 500) (3000, 0)

Looked at in normal form, there are eight possible outcomes: (A, (Enter, Enter)), (A, (Enter, Don’tEnter)), (A, (Don’t Enter, Enter)), (A, (Don’t Enter, Don’t Enter)) (B, (Enter, Enter)), (B, (Enter, Don’tEnter)), (B, (Don’t Enter, Enter)), (B, (Don’t Enter, Don’t Enter)).

Of course, some of these outcomes lead to the same final nodes in the extensive form game tree. Forexample, (A, (Enter, Enter)) and (A, (Enter, Don’t Enter)) lead to the same final node. But from thestrategic perspective, these outcomes are different because the strategies which produce them aredifferent.

E. Equilibria for Extensive From Games

1. Nash equilibria for extensive form games written in normal form

Since we have reduced the extensive form game to a normal form game, we can try to find the Nashequilibria for the game and see if they provide a sensible solution. For the example game, the point (A,(D, E) is a Nash equilibria. If the incumbent chooses technology A, then the entrant is indifferentbetween (D, E) and (D, D) but prefers these to other points. If the entrant chooses (D, E) then theincumbent prefers technology A. This is the only Nash equilibrium in the normal form game andrepresents an equilibrium of the extensive form game.

2. Multiple Nash equilibria for extensive form games written in normal form.

Many games written in normal form have more than one Nash equilibrium. If the original game is innormal form then it may be difficult to choose between these equilibria. In many cases, in if theunderlying game is one in extensive form, in which there is a sequential order of play, then we may beable to select one of these equilibria as being more likely.

Consider the following simple example where the incumbent has a monopoly position in the market. Given this position and no rivals, the incumbent makes $100.00. The entrant obviously makes zero inthis case. If the potential rival enters the market, the incumbent can follow one of two strategies:accommodate or fight. If the incumbent engages in a costly price war its returns decrease to $30.00 andthe entrant loses $15.00. If it accommodates the rival and acts as a duopolist, the returns to the incumbentare $60.00 while the entrant will make $30.00. This can be modeled as a two stage game in which theentrant moves first and the incumbent then chooses to accommodate or fight. The game tree in extensiveform is as follows.

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Entrant

Enter

Stay Out

Incumbent

Fight

Accomodate

(-15 ,30)

(30, 60)

(0, 100)

Game Tree Entry Deterrence II

E1

I1

T3

T1

T2

As before we can also write this game in normal form. The payoff matrix is as below.

Table 9 Outcomes for Entry Game II in Normal Form

Incumbent’s

Fight Accommodate

Enter (-15, 30) (30, 60)

Entrant’s Strategy

Stay Out (0, 100) (0, 100)

We can find the Nash equilibria as usual. There are two Nash equilibria to this normal form game (Enter,Accommodate) and (Stay Out, Fight). This is clear since if the entrant chooses to enter, the best strategythe incumbent is to accommodate, while if the incumbent is going to accommodate, the best strategy forthe rival is to enter. Similarly if the entrant chooses to stay out, the incumbent may as well fight asaccommodate, while if the incumbent is going to fight, the incumbent is better off to stay out. Thus thisgame seems to have two Nash equilibria.

However, the outcome (Stay Out, Fight) is not really a sensible prediction for the game, since the entrantcan foresee that if it does enter, the incumbent will in fact accommodate. Thus the concept of Nashequilibrium seems to be lacking in this game.

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3. Threats (credible and incredible)

a. When one player attempts to get other players to believe it will employ a specific strategy, itis called a threat.

In the entry deterrence game II, the incumbent would like the entrant to believe that it willfight if the entrant chooses to get in the market.

b. A threat by a player is not credible unless it is in the player’s own interest to carry out thethreat when given the option.

c. Threats that are not credible are termed incredible.

In the entry deterrence II game, the threat to fight if the rival enters the market is incredible.

d. We may be able to eliminate some Nash equilibria as being unreasonable predictions for agame if we can show they depend on incredible threats.

Consider as an example the game played between a couple, John and Mary. They areplanning to go out on a Friday evening and John needs to make reservations at a restaurantsince Mary will not have access to a telephone on her job as a wildlife biologist. John prefersthe local Sport’s Grill while Mary prefers the new Thai restaurant. Before leaving for workthey engage in a somewhat heated discussion and Mary says, “If you make the reservation atthe Sport’s Grill, I’ll just stay home with the four young children and watch cartoons and youcan go out yourself.” Later that day at work, John must decide which restaurant to call. Written in normal form this game would have 2 Nash equilibria where John and Mary bothend up at one restaurant or the other since neither prefers Friday night at home. However, wecan predict the outcome in the extensive form, since Mary’s threat to just stay home is notcredible when John gets to move first. Once a reservation has been made, Mary prefers to gothe Sport’s Grille as opposed to staying home. Knowing this, John will not make a reservationat the Thai restaurant (assuming that getting Mary upset doesn’t affect his payoff, in whichcase he may choose differently).

4. Subgames

A subgame is a decision node from the original game along with the decision nodes and terminal nodesdirectly following this node. A subgame is called a proper subgame if it differs from the original game.

Entry deterrence I has three subgames: the game itself and two proper subgames beginning with nodes E1 and E2. Entry deterrence II has two subgames: the game itself and the proper subgame with initialnode I1. Just as games may have Nash equilibria, subgames may have Nash equilibria.

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Entrant

Enter

Stay Out

Incumbent

Accomodate (30, 60)

(0, 100)

Pruned Game Tree Entry Deterrence II

E1

I1

T3

T2

5. Using subgames to refine Nash equilibria in extensive form games

a. example 1

To see how the idea of a subgame can help us predict the outcome of an extensive form game,consider entry deterrence II. Looking at the game tree, the optimal strategy for the incumbent in thesubgame is to accommodate if the potential rival enters. Thus we can eliminate the choice (fight)from this decision node. The reduced game tree looks as follows:

Having eliminated the fight option at node I1, it is clear that the best strategy for the entrant is toenter the market since 30 is greater than 0. The incumbent can threaten to fight if the rival entersbut this threat is not credible.

b. example 2

Consider now a new competition game which we denote entry deterrence III. In this game the thereis one firm in the industry and one potential rival which is considering production. The firms usethe same production technology and thus have the same cost of production per unit which weassume is constant. The larger firm is considering building a larger plant that has high fixed costsbut will have lower and constant marginal costs than the current plants. This technology is reallyopen only to the incumbent in the industry since the smaller firm does not have the distributionnetwork to effectively market the additional product. Once the large firm decides whether to investin the new plant or not , the rival decides whether to enter and each firm chooses an output level. The game tree is below.

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Incumbent

New Plant

Entrant

Entrant

Enter

Do not enter

Enter

Do not enter

(4000, -2000)

(5000, 0)

(6000, 0)

Extensive Form Game TreeEntry Deterrence III

I1

E1

E2

T1

T2

T4

T3

Older Plant

(4500, 1500)

We can also represent this game in normal form with a payoff matrix as follows.

Table 10 Outcomes for Entry Game III in Normal Form

Entrant’s Strategy

(E, E) (E, D) (D, E) (D, D)

New Plant (4000, -2000) (4000, -2000) (5000, 0) (5000, 0)

Incumbent’s Strategy

Old Plant (4500, 1500) (6000, 0) (4500, 500) (6000, 0)

We can find two Nash equilibria to this game. The first is (New Plant, (D, E)) since if theincumbent picks a new plant the best option for the rival is to stay out. But if the rival picks (D, E)the best choice for the incumbent is the new plant. The second is (Old Plant, (E, E)) since if theincumbent chooses an old plant the best strategy for the rival is to enter. But if the rival chooses toenter, the best strategy for the incumbent is to choose the old plant. The normal form gameobscures the fact that the rival can observe whether the incumbent is building the new plant.

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5. Subgame-perfect equilibrium

As an alternative to Nash equilibrium in extensive form games, a refinement know as subgame perfectequilibrium is used.

An outcome is said to be a subgame perfect equilibrium (SPE) if it induces a Nash equilibrium in everysubgame of the original game (including the game itself).

Consider now the two Nash equilibria in the entry deterrence III game. The first equilibrium is (NewPlant, (D, E)). In the subgame starting at E1, the action D is optimal for the entrant since 0 > -2000. Sothe action D is a Nash equilibrium for this simple sub-game. Similarly if we consider the subgamestarting at E2, the action E is optimal for the entrant since 1500 > 0. Thus the strategy set of the entrant(D, E) is optimal for both subgames of the original game and constitutes a subgame-perfect equilibrium.

Now consider the second equilibrium of (Old Plant, (E, E)). In the subgame starting at E1, the action Dis optimal for the entrant since 0 > -2000. So the action E is not a Nash equilibrium for this simple sub-game and the strategy (E, E) cannot be part of a subgame-perfect equilibrium of the original game.Therefore, (Old Plant, (E,E)) is not subgame-perfect..

Every subgame-perfect equilibrium is a Nash equilibrium but not every Nash equilibrium is a subgame-perfect equilibrium.

6. Backward induction

a. algorithm

In extensive games with perfect information, we can find the subgame perfect equilibrium using thefollowing technique called backward induction.

1) Start at the terminal nodes of the game tree and trace each back to its parent. Each ofthese parent nodes is a decision node for some player.

2) Find the optimal decision for that player at that decision node by comparing the rankingthe player assigns to the terminal nodes that are reached from this decision node. Recordthis choice at the node.

3) Prune from the tree all branches that originate from the decision nodes selected in step 1. The pruning now makes each of these decision nodes into a terminal node. Attach toeach of these new terminal nodes the payoff received when the optimal action is taken ateach node.

4) If this new game tree has no decision nodes, quit.

5) If the new tree still has decision nodes, return to step 1 and continue.

6) For each player collect the optimal decisions at every decision node that belongs to thatplayer. This collection of decision constitutes the player’s optimal strategy in the game.

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Incumbent

New Plant

Entrant

Entrant

Enter

Do not enter

Enter

Do not enter

(4000, -2000)

(5000, 0)

(6000, 0)

Pruned Game Tree 1Entry Deterrence III

I1

E1

E2

T1

T2

T4

T3

Older Plant

(4500, 1500)

Incumbent

New Plant

Entrant

Entrant

Do not enter

Enter

(5000, 0)

Pruned Game Tree 2Entry Deterrence III

I1

E1

E2

Older Plant(4500, 1500)

Incumbent

New Plant

(5000, 0)

Pruned Game Tree 3Entry Deterrence III

I1

b. example

For entry deterrence III consider the following set of decision trees.

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