Game Theory

1
Game Theory

• Day 2

2
Review

• Predicting likely outcome of a game
• Sequential games
• Look forward and reason back
• Simultaneous games
• Look for best replies
• What if there are multiple equilibria?
• What if there are no equilibria?

3
Mixed Strategies

• Unreasonable predictors of one-time human
  interaction
• Reasonable predictors of long-term proportions

4
Employee Monitoring

• Employees can work hard or shirk
• Salary 100K unless caught shirking
• Cost of effort 50K
• Managers can monitor or not
• Value of employee output 200K
• Profit if employee doesnt work 0
• Cost of monitoring 10K

5
Employee Monitoring

• Best replies do not correspond
• No equilibrium in pure strategies
• What do the players do?

Manager
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Mixed Strategies

• Randomize surprise the rival
• Mixed Strategy
• Specifies that an actual move be chosen randomly
  from the set of pure strategies with some
  specific probabilities.
• Nash Equilibrium in Mixed Strategies
• A probability distribution for each player
• The distributions are mutual best responses to
  one another in the sense of expectations

7
Finding Mixed Strategies

• Suppose
• Employee chooses (shirk, work) with probabilities
  (p,1-p)
• Manager chooses (monitor, no monitor) with
  probabilities (q,1-q)
• Find expected payoffs for each player
• Use these to calculate best responses

8
Employees Payoff

• First, find employees expected payoff from each
  pure strategy
• If employee works receives 50
• Profit (work) 50 ?q 50 ?(1-q)
• 50
• If employee shirks receives 0 or 100
• Profit (shirk) 0 ? q 100 ? (1-q)
• 100 100q

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Employees Best Response

• Next, calculate the best strategy for possible
  strategies of the opponent
• For qlt1/2 SHIRK
• Profit (shirk) 100-100q gt 50 Profit (work)
• For qgt1/2 WORK
• Profit shirk) 100-100q lt 50 Profit (work)
• For q1/2 INDIFFERENT
• Profit (shirk) 100-100q 50 Profit (work)

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Cycles
1
shirk
p
1/10
work
0
0
1
1/2
q
monitor
no monitor
11
Mutual Best Replies
1
shirk
p
1/10
work
0
0
1
1/2
q
monitor
no monitor
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Mixed Strategy Equilibrium

• Employees shirk with probability 1/10
• Managers monitor with probability ½
• Expected payoff to employee
• chance of each of four outcomes x payoff
  from each
• Expected payoff to manager

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Properties of Equilibrium

• Both players are indifferent between any mixture
  over their strategies
• e.g. employee
• If shirk
• If work
• Regardless of what employee does, expected payoff
  is the same

14
Use Indifference to Solve I

• 50q50(1-q) 0q100(1-q)
• 50 100-100q
• 50 100q
• q 1/2

15
Use Indifference to Solve II

• 90(1-p)-10p 100(1-p)-100p
• 90-100p 100 200p
• 100p 10
• p 1/10

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Indifference
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Why Do We Mix?

• Since a player does not care what mixture she
  uses, she picks the mixture that will make her
  opponent indifferent!

COMMANDMENT Use the mixed strategy that keeps
your opponent guessing.
18
New York Times5 November 2002

• IRS Commissioner Charles Rossotti
• Audits more expensive now than in 97
• Number of audits decreased slightly
• Offshore evasion alone increased to 70 billion
  dollars!
• Recommends
• As audits get more expensive, need to increase
  budget to keep number of audits constant!

19
Does It Work In Real Life?

• Sports
• Football
• Tennis
• Baseball
• Law Enforcement (speed control)
• Policy Compliance (check randomly)

20
Law Enforcement

• Motivate compliance at lower monitoring cost
• Audits
• Drug Testing
• Parking
• Punishment should fit the crime?

21
Law Enforcement

• Increase in cost to one party should not change
  the optimal mix for that party
• Consulting Firm Not surprisingly, it was not
  intuitive to them that a 10 increase in
  enforcement costs should not be met with a
  decrease in enforcement. By reducing the
  ticketing rate, incidence of illegal parking
  increased by over 40.

22
Caveat

• Managers strategy of monitor ½ the time must
  mean that there is a 50 chance of monitoring in
  every round!
• Cannot just monitor every other day.
• Humans are bad at this!
• Exploit patterns!

23
More Review

• Simultaneous games
• Put yourself in your opponents shoes
• Iterative reasoning
• Sequential games
• Look forward and reason back
• Sequentially rational reasoning
• New If the rules are not fixed, change them to
  benefit you.

24
Before the Supreme CourtGore v. Bush

• Four judges prefer
• Bush to Gore to Remand (BgtGgtR)
• Three judges prefer
• Gore to Remand to Bush (GgtRgtB)
• Two judges prefer
• Remand to Bush to Gore (RgtBgtG)
• No majority in three-way race
• Chief justice selects voting order
• B v. G then winner v. R
• B v. R then winner v. G
• G v. R then winner v. B

25
Changing the Rules

• Are the rules of the game fixed?
• Example voting and agenda setting

COMMANDMENT When the rules of the game are
flexible manipulate them to your advantage.
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Member Preferences

• Preferences
• 4 B gt G gt R
• 3 G gt R gt B
• 2 R gt B gt G
• Majority rule results
• B beats G G beats R R beats B

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7
5
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Rules of Voting

• Majority rule results
• B beats G G beats R R beats B
• B vs. G then winner vs. R ? R
• B vs. R then winner vs. G ? G
• G vs. R then winner vs. B ? B

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Solving Sequential Games

• Start with the last move in the game
• Determine what that player will do
• Trim the tree
• Eliminate the dominated strategies
• This results in a simpler game
• Repeat the procedure

29
Sequential Rationality
COMMANDMENT Look forward and reason
back. Anticipate what your rivals will do
tomorrow in response to your actions today
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Voting Revisited

• Majority rule results
• B beats G G beats R R beats B
• What if you want Remand to Win?
• B vs. G then winner vs. R ? R
• Problem
• Everyone knows you want R
• B vs. G then winner vs. R? Good Luck!
• Better chance
• R vs. G, then winner versus B

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Extensive Form
B
B
B vs. R
R
R
R vs. G
R
G
B
B
B vs. G
G
G
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Looking Forward
B
B
B vs. R
A majority prefers R to B
R
R
B
B
A majority prefers B to G
B vs. G
G
G
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Trim The Tree
B vs. R
R
R
R vs. G
R
G
B
B
B vs. G
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Reasoning Back

• First stage, in effect vote between R B
• Gore supporters prefer GgtRgtB, vote R

B vs. R
R
R
R vs. G
R
G
B
B
B vs. G
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What Happened?

• Gore supporters have preferences
• Gore gt Remand gt Bush
• In first round, vote between R and G
• Gore supporters prefer Gore
• But vote for G is in effect a vote for B!
• So Gore supporters vote for remand.
• Guarantee themselves second best choice

36
Rollback in Voting and Being
Political

• Not necessarily good to vote your true
  preferences
• Amendments to make bad bills worse
• Crossing over in open primaries
• Centrist voting in primaries
• Supporting your second-best option
• STILL Outcome predetermined
• AGENDA SETTING!

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Prisoners Dilemma
Pricing
Equilibrium 20K
Cooperation 25K
38
Pricing
Contribution
Reichard knows Bojangles pricing decision.
212,750166,800
Reichard
3
3
Bojangles
4
332,000 62,500
198,000258,300
3
4
Price leader
4
266,800211,200
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Repeated Interaction

• Repeated Interaction
• Ongoing relationship between players
• Current action affects future interactions
• History-Dependent Strategies
• Choose an action today dependent on the history
  of interaction
• Can history-dependent strategies help enforce
  mutual cooperation?
• It depends

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Finite RepetitionSilly Trickery

• Suppose the market relationship lasts for only T
  periods
• Use backward induction (rollback)
• Tth period no incentive to cooperate
• No future loss to worry about in last period
• T-1th period no incentive to cooperate
• No cooperation in Tth period in any case
• No opportunity cost to cheating in period T-1
• Unraveling logic goes back to period 1

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Finite Repetition

• Cooperation is impossible if the relationship
  between players is for a fixed and known length
  of time.
• Why do people cooperate even though they dont
  live forever?

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Infinite Repetition

• No last period, so no rollback
• Use history-dependent strategies
• Trigger strategies
• Begin by cooperating
• Cooperate as long as the rivals do
• Upon observing a defection immediately revert
  to a period of punishment of specified length in
  which everyone plays non-cooperatively

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Two Trigger Strategies

• Grim Trigger Strategy
• Cooperate until a rival deviates
• Once a deviation occurs, play non-cooperatively
  for the rest of the game
• Tit-for-Tat Strategy
• Cooperate if your rival cooperated in the most
  recent period
• Cheat if your rival cheated in the most recent
  period

44
Grim Trigger Strategy

• In any period t, a firm faces one of two
  histories of play
• Zero deviations up to that point
• Charge the high price in the next period
• One or more deviations up to that point
• Charge the low price from that point on in every
  period
• Since low, low is the Nash equilibrium, each
  firm is doing the best it can

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Equilibrium in GTS Discounting

• Discounting value of future profits is less than
  value of current profits
• r is the interest rate
• is the discount factor
• Invest
• 1 today get (1r) tomorrow
• ? today, get 1 tomorrow

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Payoff Stream
profit
28
collude
25
cheat
20
t
t1
t2
t3
time
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Sustainability

• The minimum discount rate required to sustain the
  collusive outcome depends on the payoff structure
• Greater relative profits from cheating
• Need lower interest rate (larger discount factor)
• Smaller relative profits after cheating
• Need higher interest rate (smaller discount
  factor)

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Tit-for-Tat

• Tit-for-Tat is nicer than GTS
• If rival uses tit-for-tat, cooperate if
• a) Colluding is better than cheating
• 252525 gt 28 20 202020
• b) Colluding is better than cheating once
• 252525 gt 28 15 252525

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Axelrods Simulation

• R. Axelrod, The Evolution of Cooperation
• Prisoners Dilemma repeated 200 times
• Economists submitted strategies
• Pairs of strategies competed
• Winner Tit-for-Tat
• Reasons
• Forgiving, Nice, Provocable, Clear

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Main Ideas

• Not necessarily tit-for-tat
• Doesnt always work
• Dont be envious
• Dont be the first to cheat
• Reciprocate opponents behavior
• cooperation and defection
• Dont be too clever

51
Trigger Strategies

• GTS and Tit-for-Tat are extremes
• Two goals
• Deterrence
• GTS is adequate punishment
• Tit-for-tat might be too little
• Credibility
• GTS hurts the punisher too much
• Tit-for-tat is credible

52
Inducing Cooperation

• Trigger strategies
• Announce the trigger
• Announce the punishment

COMMANDMENT In announcing a punishment
strategy Punish enough to deter your opponent
Temper punishment to remain credible
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Conclusion

• Cooperation
• Struggle between high profits today and a lasting
  relationship into the future
• Deterrence
• A clear, provocable policy of punishment
• Credibility
• Must incorporate forgiveness

54
Game Theory

• The Power to Constrain an
  Adversary Depends Upon the Power to Bind
  Oneself.
• - Thomas Schelling