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Game Theory. Games of strategy Sequential games Simultaneous decisions Dominated strategies Nash equilibrium Prisoners’ dilemma. Sequential decisions. Previously …. Sequential decisions with uncertainty Decision trees … with “chance” nodes but …

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game theory

Game Theory

Games of strategy

Sequential games

Simultaneous decisions

Dominated strategies

Nash equilibrium

Prisoners’ dilemma

sequential decisions
Sequential decisions
  • Previously ….
  • Sequential decisions with uncertainty
  • Decision trees … with “chance” nodes
  • but …
  • “God does not play dice” – Albert Einstein
  • “Subtle is the Lord, but malicious He is not.”
  • What about your competitors?
a sequential game
A sequential “game”
  • Decisions made in sequence.
  • Your decision depends on decision made previously by others, and others’ decisions follow and depend on yours, etc.
  • Outcome/payoff depends on all decisions made by all.
lucy van pelt vs charlie brown
Lucy Van Pelt vs. Charlie Brown
  • Lucy Van Pelt holds a football on the ground and invites Charlie Brown to run up and kick it. At the last moment, Lucy pulls the ball away. Charles Brown, kicking air, lands on his back, and this gives Lucy great perverse pleasure.
representing decisions in a game tree

Pull Ball Away

Accept

Lucy

Charlie

Let Charlie kick

Reject

Representing Decisions in a Game Tree

,

,

,

games of strategy
Games of Strategy

Vijay Krishna (Harvard Business School):

Any situation where the choices of two or more rational decision makers together leads to gains and losses for them is called a game.

A game may simultaneously involve elements of both conflict and co-operation among the decision makers.

market competition hdtv
Market Competition - HDTV

Vizio considers entering a market now monopolised by Samsung. Samsung can decide to respond by being accommodating or aggressively fight a price war. Profit outcomes for both firms depends on the strategies of both firms.

As Vizio, you can analyse this problem using Decision Analysis by estimating probabilities of Samsung’s responses.

market entry decision tree for vizio
Market Entry – Decision Tree for Vizio

Accommodate

$100,000 to Vizio

p

Samsung

Enter

1-p

-$200,000 to Vizio

Vizio

Fight price war

Keep out

$0 to Vizio

How to estimate the probabilities? What does p depend on?

If no information, p=0.5? Then Vizio will not enter market.

game tree representation
Game Tree Representation

5, 8

Accommodate

Probability of Samsung’s response will depend on Samsung’s payoff in the different scenarios

Samsung

Enter Market

Aggressive

Vizio

-7, 2

Do not enter

0,10

market entry game tree model
Market Entry – Game Tree Model

$100,000 to Vizio

$100,000 to Samsung

Accommodate

Samsung

-$200,000 to Vizio

-$100,000 to Samsung

Enter

Vizio

Fight price war

Keep out

$0 to Vizio

$300,000 to Samsung

analysing game trees
Analysing Game Trees

Rule 1: Look Ahead and Reason Back!

  • For this market alone, Vizio should choose enter because Samsung (rationally) will accommodate.
  • If Samsung worries that Vizio may enter other markets in the region after this, Samsung may take a tough stand. Vizio should not enter.
  • The “payoff” should include all “benefits”.
look ahead reason back
Look Ahead & Reason Back
  • Formulate the game tree of the situation.
    • Identify your own and opponent’s strategy at each stage. This assumes:
      • Your opponent’s strategy can be observable.
      • Strategy must be irreversible.
  • Evaluate payoffs at the “leaves” of the tree.
    • Think about what will happen at the end.
  • Reason backward through the tree.
    • Identify the best strategy for each player at each stage, starting at the end.

Note the essence of a game of strategy is interdependence. Your decision affects your opponent’s decision and your opponent’s decision affects yours.

more complex games

Black-1

Black-1

Black-1

White-2

White-2

White-2

More complex games

P-Q4

P-QB4

P-K4

White-1

Theoretically, can map out all possible chess moves and then select the best sequence of moves to win the game!

chess human vs computers
Chess - Human vs. Computers
  • Good chess players can “see” 14 moves ahead!
  • (1968) David Levy: “No computer can beat him in 10 years”
  • Deep Blue
    • Chess playing machine built by IBM in the 1990’s
    • 2 to 2.5 million moves per second.
  • (1996) Deep Blue 1 lost to world chess champion Gary Kasparov.
  • (1997) Deep Blue 2 defeated Kasparov.
homework
Homework?
  • Draw the game tree for TIC-TAC-TOE.
  • Sure-win strategies?
the election of the chief executive for hong kong
The Election of the Chief Executive for Hong Kong
  • The CE of Hong Kong SAR Government Mr. Donald Tsang finishes his term in 2012, and the next CE will be “elected”.
  • Mr. Henry Tang is Beijing’s favourite candidate.
  • Mr. Alan Leung (the potential challenger) considers entering the race.
  • Tang must determine whether to launch a preemptive advertising campaign against Leung (expensive) or not (cost-saving).
  • Leung must determine whether to enter the race.
election game tree
Election Game Tree

Tang’s, Leung’s payoff

In

1, 1

Leung

The larger the better

Advertise

Out

3, 2

Tang

In

2, 4

No Ad

Leung

Out

4, 2

game tree
Game Tree

Tang’s, Leung’s payoff

In

1, 1

Leung

Ads

Out

3, 2

Tang

In

2, 4

No Ads

Leung

Out

4, 2

game tree22
Game Tree

Tang’s, Leung’s payoff

In

1, 1

Leung

Ads

Out

3, 2

Tang

In

2, 4

No Ads

Leung

Out

4, 2

advantage due to order of decisions
Advantage due to Order of Decisions?
  • First-mover advantage?
  • Tang (first) sets the stage for Leung (second). Tang can look ahead to Leung’s optimal response and make the move to his advantage.
  • Can Leung improve his situation by acting first?
game tree24
Game Tree

Leung’s, Tang’s payoff

Adv

1, 1

Tang

In

No Adv

4, 2

Leung

Adv

2, 3

Out

Tang

No Adv

2, 4

better off being first
Better off being first?
  • Is there a first-mover advantage?
  • What about adoption of new technology?
    • Better off as a technology leader?
    • Better off as a technology follower?
simultaneous decisions

Simultaneous Decisions

In the chess example, the sequence of decisions alternate between the players.

In other situations, the decision may not be sequential but simultaneous.

Tic-tac-toe (sequential)

Stone-paper-scissors (simultaneous)

In simultaneous games, the payoffs to the players are still interdependent on chosen strategies of ALL players.

time vs newsweek
Time vs. Newsweek
  • Each week, these magazines decide on what story to put on the cover.
  • They do not know the other’s decision until publication.
  • Suppose there are two “hot” stories:
    • (A): Anna Chapman, the Russian Spy,
    • (B): British Petroleum Oil Spill damage
  • Newsstand buyers only purchase if story is on cover.
  • 70% interested in (A) and 30% in (B).
    • Purchases evenly split the if both magazines have the same story.
matrix representation of game
Matrix Representation of Game

What should Time do?

matrix representation of game for newsweek
Matrix Representation of Game for Newsweek

No matter what Time does, Newsweek is better off putting (A) as cover story.

dominant strategies
Dominant Strategies

Choosing (A) is a dominant strategy for both Time and Newsweek!

dominant strategy
Dominant Strategy
  • A dominant strategy is one that makes a player better off than he would be if he used any other strategy, no matter what strategy his opponent uses.
  • A strategy is dominated if there is another strategy that under no circumstances leads to a lower payoff, and sometimes yields a better payoff.
  • Note: For some games, there may be no dominant strategy for some players.
properties of a dominant strategy
Properties of a dominant strategy

1: A dominant strategy dominates your other strategies, NOT your opponent!

Even with your dominant strategy, your payoff could be smaller than your opponents.

2: A dominant strategy does not requires that the worst possible outcome of the dominant strategy is better than the best outcome of an alternative strategy.

pricing example
Pricing example

Suppose there are just two possible pricing choices: $3 (a profit margin of $2 per copy) and $2 ($1 per copy). Customers will always buy the lower-priced magazine. Profits are split equally between the two. The total readership is 5 million if the price is $3, and rises to 8 million if the price is only $2.

analysing games
Analysing Games

Rule 2: If you have a dominant strategy, use it!

Rule 3: Eliminate any dominated strategies from consideration, and do so successively!

eliminating dominated strategies example

C

B

F

E

A

I

D

H

G

Eliminating Dominated Strategies - Example
  • American ship at A, Iraqi ship at I.
  • Iraqi plans to fire a missile at American ship; American ship plans to fire a defense missile to neutralize the attack (simultaneously).
  • Missiles programmed to (possibly) turn every 20 seconds.
  • If missile not neutralised in 60 seconds, American ship sinks!
possible strategies paths
Possible strategies (paths)
  • For American,
    • A2, A3 dominated by A4,
    • A6, A7 dominated by A8,
    • A1 is dominated by A8,
    • A5 is dominated by A4,
    • Only A4 and A8 not dominated.
  • Similarly for Iraqi.
    • Only I1 and I5 are not dominated.
simplified game
Simplified Game

C

B

F

A

E

I

D

H

G

No dominant strategy for either player!

nash equilibrium
Nash Equilibrium
  • A set of strategies constitute a Nash Equilibrium if:

no player can benefit by changing her strategy while the other players keep their strategies unchanged.

Each player’s strategy is the “best-response” to the other players’ set of strategies.

dominant strategy equilibrium
Dominant Strategy Equilibrium
  • Higher viewership means more advertising revenues for both TV stations.
  • Each TV station has a dominant strategy.
  • In this case, the equilibrium for this game is obvious.
dominant strategy equilibrium40
Dominant Strategy Equilibrium
  • If, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of (dominant) strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.
nash equilibrium41
Nash Equilibrium

No dominant strategy for Newsweek.

Unique Nash equilibrium.

example chicken
Example - “Chicken”

H > C > D

  • No dominant strategy
  • Two Nash equilibria

James Dean

Swerve

Don’t

swerve

Mad

Swerve

C, C

C, H

Max

Don’t

H, C

D, D

swerve

choosing among multiple equilibria
Choosing among Multiple Equilibria
  • Some games have multiple equilibria.
  • “Rule of the road”
    • Hong Kong, Britain, Australia, Japan (left)
    • China, USA (right)
  • The social convention of the locale determines which equilibrium to choose.

Sweden switch from left to right in 1967.

in class exercise texas a m
In-class exercise (Texas A&M)
  • Each of you owns a production plant and can choose to produce 1 or 2 units of a product.
  • More total production will lower price and hence profit.

What would you do?

is a nash equilibrium good for the players
Is a Nash equilibrium “good” for the players?

Just because a game has an equilibrium does not mean that those strategies are “best” for the players.

  • Prisoners’ dilemma:
    • Two burglars, Bob and Al, are captured at the scene of a burglary and interrogated separately by the police.
    • Each has to choose whether or not to confess.
    • Outcomes:
      • If neither man confesses, then both will serve only one year.
      • If both confesses, both will go to prison for 10 years.
      • However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years.
prisoners dilemma
Prisoners’ dilemma

1950 – Dresher & Flood (Rand)

A. W. Tucker

  • For each player, the dominant strategy is to confess!
  • Unique Nash equilibrium!
  • Both play the dominant strategy but create mutually disastrous outcome! Both would be better off by denying!
cartels
Cartels
  • Companies or countries form an alliance to jointly make price and production decisions.
  • World Trade Organisation (WTO) / General Agreement on Tariffs and Trade (GATT)
  • The Organization of Petroleum Exporting Countries (OPEC) is a cartel.
    • the mission of OPEC is to coordinate and unify the policies of its Member Countries and ensure the stabilization of oil markets in order to secure a regular supply of petroleum to consumers, a steady income to producers …
opec maintaining a cartel
OPEC – Maintaining a Cartel
  • Total output: 4mb 6mb 8mb
  • Price per barrel: $25 $15 $10
  • Extraction costs: Iran: $2/barrel;

Iraq: $4/barrel

Dominant strategy: produce at higher level !

ensuring co operation
Ensuring Co-operation
  • The dominant strategy equilibrium results in each producing 4 million barrels and achieving 56 million in total joint profit.
  • Suppose OPEC countries have agreed to maintain production at 2 mb per day.
  • If members produces 2 million barrels each (as agreed), they will make 88 million in total joint profit.
  • Is it possible to achieve cooperation, when the dominant strategy is to cheat?
detection of cheating
Detection of Cheating
  • Co-operation is difficult when the reward for cheating is high.
  • How to tell if some member cheated and produced more?
  • The price is US$25 per barrel only if members maintained low production. If price drops below $25, then someone has cheated!
  • What if demand actually decreased?
identifying cheaters
Identifying cheaters
  • In a two-player game, an honest party knows who cheated.
    • Still, the cheating party may deny they cheated.
  • When there are many players, even when cheating has been detected, it may be difficult to identify who cheated !
  • If voluntary cooperation is not possible, how about making use of punishment?
    • In a one-period game, there is no solution to achieve reciprocal co-operation.
punishment credible threat
Punishment – Credible Threat?
  • If the game repeats, cooperation may be enforced.
  • Suppose Iran begins to cheat and produces 4 million barrel per day secretly.
    • Iran’s profit goes up from 46 to 52 million per day.
  • When Iraq finds out, Iraq also produces 4 million barrels.
    • Iran’s profit goes down to 46 to 32 million per day.
  • Assume it takes a month for Iraq to know.
    • Iran’s total profit through cheating: 6x30= 180 million
  • Iraq retaliates by increasing production.
    • Iran’s cheating gain will be wiped out in 13 days (i.e., 180 million / 14 million)
competition or collusion
Competition or Collusion?
  • DVD player vendors: Fortress

Broadway

  • wholesale: $1500, retail: $3000
  • Broadway: lowest price guarantee:

“refund double the price difference”

  • Should Fortress cut its price to $2750?
  • What will the consumer do?
  • How will Broadway respond?
implicit cartel
“Implicit” Cartel
  • If Fortress tries to increase its market share by lowering its price to $2750.
  • Customers will buy from Broadway and claim from a $500 rebate. The “selling price” for Broadway is effectively $2500; lower than Fortress’ price of $2750.
  • In response, Broadway will not give away rebates but lower its price to $2750.
  • Fortress becomes worse off … so why bother?

Collusion is enforced by “announcing” the punishment!

sustaining co operation
Sustaining Co-operation
  • Mechanism must Detect cheating and Deter cheating.
  • Which Punishment?
  • Simplicity and clarity
    • Easy for potential cheaters to evaluate consequences.
  • Certainty
    • Players have confidence that defection will be punished and co-operation rewarded.
  • Severity
    • not to “fit the crime” but for deterrence!

Risk of mistakes?

tit for tat
Tit-for-tat

Exodus 21:22-25

  • If men who are fighting hit a pregnant woman and she gives birth prematurely but there is no serious injury, the offender must be fined whatever the woman’s husband demands.
  • But if there is a serious injury, you are to take life for a life, eye for eye, tooth for tooth, hand for hand, burn for burn, wound for wound, bruise for bruise.
tit for tat strategy
Tit-for-tat Strategy

Co-operates in the first period, thereafter mimics the rival’s action from previous rounds

  • Clarity (simple to implement)
  • Niceness (does not initiates cheating)
  • Provocability (it never lets cheating go unpunished)
  • Forgiveness (does not hold a grudge, willing to restore cooperation)
  • “Chain reaction” of mistakes?
hatfields and mccoys
Hatfields and McCoys
  • This is one of best-documented stories on inter-family feud (1878 – 1891).
  • Early settlers in the Tug Valley on the Kentucky and West Virginia border.
  • Feud started over the disputed ownership of a pig!

West Virginia

Kentucky

tit for tat with misperception
Tit-for-tat with misperception

Mis-perception leads to perpetual retaliation!

Nuclear conflict?

Cuban missile crisis (1962).

tit for tat strategy60
Tit-for-tat Strategy
  • When misperceptions are possible, in the long run tit-for-tat will spend half the time cooperating and half of it defecting.
    • When the probability of a misperception is small, it will take a lot longer for this phenomenon to occur.
    • When the probability is 50%, whatever you do will not have any affect on your opponent.
        • Opponent will perceive aggression with 0.5 probability.
  • When the probability is 50%, there is no hope of achieving co-operation. One should always attack!
  • Feud never ends …

Should one be more forgiving?

the moderate chinese way
中庸之道 (The Moderate Chinese Way)
  • Tit-for-tat is quick to punish opponent who has a long history of cooperation.
  • Other responses:
  • (Matthew 5:38): “But I tell you, do not resist an evil person. If someone strikes you on the right cheek, turn to him the other also.”
  • A more forgiving tit-for-tat:
    • Begin cooperating
    • Continue cooperating, but keep count of how many times the other side appears to be have defected while you have cooperated.
    • If the this percentage becomes unacceptable, revert to tit-for-tat.
summary
Summary
  • Games of strategy
  • Sequential games
  • Simultaneous decisions
  • Dominated strategies
  • Nash equilibrium
  • Prisoners’ dilemma
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