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Chapter 5. Stable Play: Nash Equilibrium in
Discrete n-Player Games
Prepared by Philip Heap
Revised and Presented by Shuting He
Copyright © 2014 by Worth Publishers
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5.1 Introduction
 In this chapter, you will examine games in which there are more
than three players.
 You will also look at games in which there are an unspecified
number of players, n.
 In a n-player game, because of the dimensionality limit of tabular
representation and uncertainty of values of n, the game can not be
represented in a payoff matrix.
 In some games, players will all act the same. In other games,
players act differently.
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5.1 Introduction
 Two forces to keep in mind:
 Tipping
 The tendency for a player to be more attracted to a strategy, the
more players choose it (fashion choices).
 Congestion
 The tendency for a player to be less attracted to a strategy, the
more players use it (driving routes).
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5.2 Symmetric Games
 A game is symmetric when:
1. All players have the same strategy sets.
2. Players receive the same payoff when they choose the same
strategy.
3. If you switch two players’ strategies, then their payoffs
switch as well.
So if one player’s strategy is a best reply, then it must be a best
reply for the other players.
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5.2 Symmetric Games
1. Both players can choose from Low, Moderate, or High.
2. When they choose the same strategy, the payoffs are the same.
3. If you switch their strategies, their payoffs switch.
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5.2 Symmetric Games
 Even if we change the strategies for Player 2, the game is still
symmetric.
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5.2 Symmetric Games
 What are the Nash equilibria for this game?
 This game illustrates a useful property.
 In a symmetric game, if a strategy profile is a Nash equilibrium,
then so is a strategy profile that has players swap strategies.
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5.2 Symmetric Games
 In a n-player game and symmetric game, if a strategy profile is a
Nash equilibrium, then so is a strategy profile that has two or
more players swap strategies.
 That is, in a two-player symmetric game, if (s', s'') is a Nash
equilibrium, then so is (s'', s').
 And in a symmetric three-player game, if (s', s'', s''') is a Nash
equilibrium, then so are (s', s''', s'''), (s'', s', s'''), (s''', s'', s'),
 (s''', s', s''), and (s''', s'', s').
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External: Guess & Verify
 One technique for finding the Nash equilibria is to guess and
verify using the stability of Nash equilibrium.
 First, if you recognize the tipping or congestion forces or use your
intuition of the game, you are able to form a reasonable guess of
the Nash equilibria.
 Then, for each proposed solution, you need to verify whether it is
stable, i.e. does everyone finds unilateral deviation is worse off or
as good as staying in this solution point.
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External: Stag Hunt
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External: Stag Hunt
1,0
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External: Stag Hunt
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External: Stag Hunt
 Suppose there are n hunters, the split payoff of a stag for each
hunter is 2 (i.e. large enough for all hunters to share) and the
payoff of a hare is 1
 First, make a reasonable guess of Nash equilibrium: based on the
tipping incentive of all players, NEs could be (Stag,…., Stag) and
(Hare,…, Hare).
 Then, let’s verify it together.
 Let’s suppose (n/2 hunters go for Hare, n/2 hunters go for Stag) is
a NE and verify it is not.
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External: Stag Hunt
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Operating Systems: Mac or Windows
 How does a single product come to dominate a market?
 Network effects – the value of a product to a consumer is greater
when more consumers use it. Therefore, we see a tipping force.
 Suppose there are n ≥ 2 people who can buy a Mac or a PC.
 Payoff from a Mac = 100 + 10 × m
 Payoff from a PC = 10 × w
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Operating Systems: Mac or Windows
 Assume everyone buys so w + m = n
 What are the Nash equilibria?
1. Every consumer buys the superior Mac.



Given (n – 1) buy a Mac, the nth consumer buys a Mac if:
Payoff from a Mac
>
Payoff from Windows
100 + 10 × n
>
10 × 1
(which holds, since n > 2 by definition)
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Operating Systems: Mac or Windows
2. Every consumer buys a PC when there are at least 11 consumers





Given (n – 1) buy a PC, the nth consumer buys a PC if
Payoff from Windows ≥
Payoff from a Mac
10 × n
≥
100 × 10 × 1
n
≥
11
So if n > (=) 11, the payoff from Windows is > (=) to payoff
of a Mac.
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Operating Systems: Mac or Windows
 So there are two Nash equilibria:


Everyone uses a Mac.

If n ≥ 11, everyone use Windows.
Network effects and tipping.
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Operating Systems: Mac or Windows n=20
100 + 10 × (20-w)
Suppose 15/19 use W.
What does the last
consumer buy?
Payoff from W = 160
Payoff from M = 150
10 × w
So she buys W.
What if 16/19 use W?
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Operating Systems: Mac or Windows n=20
100 + 10 × (20-w)
Suppose 14/19 use W.
What does the last
consumer buy?
Payoff from W = 150
Payoff from M = 160
10 × w
So she buys M.
What if 13/19 use W?
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Operating Systems: Mac or Windows n=20
 So the tipping point is 15.
 As long as a consumer expects 15 or more people to buy
Windows, she will buy Windows.
 As long as a consumer expects 14 or less people to buy
Windows, she will buy Mac.
 Consumers’ expectations tend to be self-fulfilling.
 How does a company convince potential consumers that its
product will be the popular one?
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Applying for an Internship
 Two possible choices: JPM and LM.
 Everyone prefers a JPM internship to an LM internship.
 JPM has one position. LM has three.
 You can apply for only one internship.
 Ten students from your school are applying.
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Applying for an Internship
 If a student is assured of getting an internship at JPM, his payoff
= 200. He must be the only applicant.
 If a student is assured of getting an internship at LM, his payoff =
100. He must be one of three applicants.
 The chance of getting the applied position declines as more
students apply for the same position, so does expected payoffs. So
here we see a congestion force.
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Applying for an Internship
Show that 4 JPM and 6 LM is a Nash equilibrium, and that no student has incentive
to deviate.
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Applying for an Internship
Show that 4 JPM
and 6 LM is a Nash
equilibrium.
Show that a student
cannot do better by
switching.
Right now applicants
get a payoff of 50.
If a JPM student
switched to LM, their
payoff falls to 45.
If an LM student
switched to JPM, their
payoff falls to 40.
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Applying for an Internship
Show that 4 JPM
and 6 LM is a Nash
equilibrium.
Since a student
cannot do better by
switching, 4 JPM and
6 LM is a Nash
equilibrium.
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External: Payoff Function
 Suppose the winner of an internship position is randomly chosen
among the applicants if the number of applicants exceeds the
number of internship available in each company.
 Suppose the n is large enough, so that we only need to consider
the situation the number of applicants exceeds the number of
internship available in both companies.
 First, based on the congestion force, find the Nash equilibrium.
 Let’s verify the Nash equilibrium.
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Mac or Windows vs. JMP or LM Internship
 What’s the difference between the two games?
 In Mac/Windows, the more consumers who choose one system,
the greater the payoff. Same in the Stag Hunt game.
 In JMP/LM, the more students who apply for one internship,
the lower the payoff.
 So in one game there is tipping; in the other, congestion.
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Airline Security
 A weakest link coordination game.
 There are n ≥ 2 airlines.
 Each has a strategy set [1, 2, 3, 4, 5, 6, 7]. Each number
represents a level of security expenditure by an airline.
 si denotes the strategy of airline i.
 The cost associated with a security measure is 10 × si.
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Airline Security
 The overall level of security is determined by the “weakest
link”:
 50 + 20 × min{s1, … , sn}
 Therefore, the payoff to airline i is
 50 + 20 × min{s1, … , sn} – 10 si
 Find the Nash equilibria.
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Airline Security
 First, realize that if any Nash equilibria exist, they must be
symmetric—all airlines choose the same security level.
 Suppose
> min{ , . . . ,
lowest security expenditure.
}, so that airline i doesn’t have the
 Then if airline i reduces its expenditure by one unit (from
1), the overall security level is unaffected, min{
unchanged.
,...,
to }, is
 However, the airline’s cost is now reduced by 10 from 10 ×
10 × (
- 1),
to
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Airline Security
 First, realize that if any Nash equilibria exist, they must be
symmetric—all airlines choose the same security level.
 Let s′ denote the common strategy, and focus on airline 1.
 Airline 1’s payoff is:
 = 50 + 20 × min{s′, … , s′} – 10 × s′
 = 50 + 10 s′
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Airline Security
 To see whether s′ is optimal for Airline 1, you need to see
whether they have an incentive to change their security level.
 Suppose Airline 1 chooses more security, s″ > s′. Its payoff is:
 = 50 + 20 × min{s″, s′, … , s′} – 10 × s″
 = 50 + 20 s′ – 10 s″ < 50 + 10 s′ – 10(s″ – s′)
 Therefore, Airline 1 would not want to increase security.
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Airline Security
 Do they have an incentive to reduce their security level?
 Suppose Airline 1 chooses weaker security, s0 < s′. Its payoff is:
 = 50 + 20 × min{s0, s′, … , s′} – 10 × s0
 = 50 + 20 × s0 – 10 × s0 = 50 + 10 × s0 < 50 + 10 × s′
 Therefore, Airline 1 would not want to reduce security.
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Airline Security
 The same analysis holds for all airlines.
 Therefore, all airlines choosing a security level s′ is a Nash
equilibrium.
 There are seven Nash equilibria, one for each of the seven
possible strategies, with all airlines choosing the same strategies.
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Airline Security
 Do the airlines prefer one equilibria to the others?
 Yes. Their payoff is 50 + 10 × s′, which is increasing in s′.
 Therefore, if they choose s′ = 7, their payoff is 120.
 This is the payoff dominant Nash equilibrium.
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A Weakest Link Experiment
 Payoffs are the same as in the Airline Security game.
 What do the results from the experiment indicate?
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5.3 Asymmetric Games
 Alternative sources of asymmetry.
 Players have different strategy sets.
 Players face the same choices but have different payoffs.
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External: Voting Game
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External: Voting Game
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External: Voting Game
Suppose a voter gets a payoff of 1 when a voter’s most preferred
candidate is selected, payoff of 0 when tied, and payoff of -1 when losing.
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External: Voting Game
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External: Voting Game
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External: Voting Game
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External: Voting Game
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External: Voting Game
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External: Voting Game
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External: Voting Game
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Civil Unrest
 How can coordination failure lead to civil unrest?
 500 citizens; each must decide whether to protest.
 Benefit to protesting is (50 × m): m is the number of protesters.
(tipping force)
 Cost of protesting depends on your type: Radical, Progressive,
Bourgeois.
 A person will protest if her net benefit is ≥ 0.
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Civil Unrest
 The critical mass is the minimum protest size necessary to induce
a person to participate.
 For a Radicals to participate:
or
 50 × m – 6,000 ≥ 0
 For a Progressive to participate:
or
 50 × m – 8,000 ≥ 0
 For a Bourgeoisie participate:
m ≥ 120
m ≥ 160
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Civil Unrest
 The critical mass is the minimum protest size necessary to induce
a person to participate.
 For a Radicals to participate:
or
 50 × m – 6,000 ≥ 0
 For a Progressive to participate:
or
 50 × m – 8,000 ≥ 0
 For a Bourgeoisie participate:
 50 × m – 20,000 ≥ 0
or
m ≥ 120
m ≥ 160
m ≥ 400
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Civil Unrest
 For a Radicals to participate:
 50 × m – 6,000 ≥ 0
or
 For a Progressive to participate:
 50 × m – 8,000 ≥ 0
or
 For a Bourgeoisie participate:
or
 50 × m – 20,000 ≥ 0
m ≥ 120
m ≥ 160
m ≥ 400
 What do you see?
 If it is optimal for a progressive to protest, then it is also
optimal for a radical to do so.
 If a bourgeois finds it optimal to protest, then so do radicals
and progressives.
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Civil Unrest
 From the tipping forces and this property, we can make reasonable
guess of Nash equilibria:
1. no one protests
2. only radicals protest
3. only radicals and progressives protest
4. everyone protests.
 Since you can rule out from this property:
only progressives protest
2. only Bourgeois protest
3. only progressives and Bourgeois protest
1.
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Civil Unrest
 Four possible Nash equilibria.
1. No one protests.
 Their payoffs are (0,0,0)
 If one Radical protests his payoff = 50 × 1 – 6,000 = –5,950
 If one Progressive protests her payoff = 50 × 1 – 8,000 = –7,950
 If one Bourgeois protests her payoff = 50 × 1 – 20,000 = –19,950
 Unilateral deviation is not better off by receiving negative net benefits
 So no one protesting is a Nash equilibrium.
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Civil Unrest
 Four possible Nash equilibria.
2. All Radicals protest.
 A Radical’s payoff = 50 × 100 – 6,000 = –1,000
 And Progressive and Bourgeois' payoffs = 0
 Their payoffs are (-1,000,0,0)
 If one Radical not protests his payoff = 0
 Radical will have a higher payoff (0) if he deviates from protesting
 So all Radicals protesting cannot be a Nash equilibrium.
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Civil Unrest
 Four possible Nash equilibria.
3. All Radicals and all Progressives protest.
 A Radical’s payoff = 50 × 200 – 6,000 = 4,000
 A Progressive’s payoff = 50 × 200 – 8,000 = 2,000
 Their payoffs are (4,000,2,000,0)
 If one Bourgeois protests her payoff = 50 × 201 – 20,000 = –9,950
 Unilateral deviation is not better off for all of them.
 So all Radicals and Progressives protesting is a Nash equilibrium.
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Civil Unrest
 Four possible Nash equilibria.
4. Everyone protests.
 A Bourgeois’ payoff = 50 × 500 – 20,000 = 5,000
 So everyone protesting is a Nash equilibrium.
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Civil Unrest
 In equilibrium there is (1) no protest, (2) a moderate protest, or (3)
a massive protest.
 As with some of the other games, expectations and beliefs matter.
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Entry Into a Market
 Suppose five firms are considering entering a new market.
 Firms simultaneously decide whether to Enter or Do Not Enter.
 Each firm faces a different entry cost.
 Each firm earns the same gross profit upon entry.
 A firm’s net profit = gross profit – its entry cost.
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Entry Into a Market
Company 1 is the most efficient;
Company 5 is the least efficient.
Gross profits fall with entry.
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Entry Into a Market
Company #1: If no other firms enter its payoff = 1,000 – 100 = 900
Company #3: If two other firms enter its payoff = 250 – 180 = 70
Company #5: If four other firms enter its payoff = 100 – 210 = –110
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Entry Into a Market
Does a Nash equilibrium exist when there are less than three firms?
No. Any of the five companies would find it profitable to enter.
Does a Nash equilibrium exist when there are more that three firms?
No. The gross profit for a company when there are four entrants is
150, but only company 1 has an entry cost that doesn’t exceed 150. If
four companies enter, then at least one of them must have a negative
payoff, which means that it is better for it to not enter.
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Entry Into a Market
Does a Nash equilibrium having three companies enter but not having
company 1 be one of them ?
No, if company 1 is not one of the three entrants, then its payoff is
zero, but it can then earn 50 by entering.
Nash equilibrium are:
Yes. (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5), and (1, 4, 5).
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Entry Into a Market
Explain why (1, 2, 4) is a Nash equilibrium.
Each earns a positive payoff (column 2), and Company 3 and 5 would
have a negative profit if they entered (column 3)
Is there a shortcut?
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Entry Into a Market
We know that if three firms enter, Company 1 must be one of them.
Any of the other four firms earn a positive payoff from entry when
two other firms enter, but a negative payoff if three other firms do.
Therefore, a Nash equilibrium has Company 1 and any other two
firms enter.
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Summary
 A symmetric game is one in which all players face the same
choice, and receive the same payoffs.
 In tipping, a strategy is more attractive when more players use it.
 Games with tipping tend to result in equilibria in which
everyone (or almost everyone) does the same thing.
 In congestion, a strategy is less attractive when more players use it.