Game Theory

1
Game Theory

• Game theory was developed by John Von Neumann and
  Oscar Morgenstern in 1944 -
• Economists!
• One of the fundamental principles of game theory,
  the idea of equilibrium strategies was developed
  by John F. Nash, Jr. (A Beautiful Mind), a
  Bluefield, WV native.
• Game theory is a way of looking at a whole range
  of human behaviors as a game.

2
Components of a Game

• Games have the following characteristics
• Players
• Rules
• Payoffs
• Based on Information
• Outcomes
• Strategies

3
Types of Games

• We classify games into several types.
• By the number of players
• By the Rules
• By the Payoff Structure
• By the Amount of Information Available to the
  players

4
Games as Defined by the Number of Players

• 1-person (or game against nature, game of chance)
  
• 2-person
• n-person( 3-person up)

5
Games as Defined by the Rules

• These determine the number of options/alternatives
  in the play of the game.
• The payoff matrix has a structure (independent of
  value) that is a function of the rules of the
  game.
• Thus many games have a 2x2 structure due to 2
  alternatives for each player.

6
Games as Defined by the Payoff Structure

• Zero-sum
• Non-zero sum
• (and occasionally Constant sum)
• Examples
• Zero-sum
• Classic games Chess, checkers, tennis, poker.
• Political Games Elections, War
• Non-zero sum
• Classic games Football (?), DD, Video games
• Political Games Policy Process

7
Games defined by information

• In games of perfect information, each player
  moves sequentially, and knows all previous moves
  by the opponent.
• Chess checkers are perfect information games
• Poker is not
• In a game of complete information, the rules are
  known from the beginning, along with all possible
  payoffs, but not necessarily chance moves

8
Strategies

• We also classify the strategies that we employ
• It is natural to suppose that one player will
  attempt to anticipate what the other player will
  do. Hence
• Minimax - to minimize the maximum loss - a
  defensive strategy
• Maximin - to maximize the minimum gain - an
  offensive strategy.

9
Iterated Play

• Games can also have sequential play which lends
  to more complex strategies.
• (Tit-for-tat - always respond in kind.
• Tat-for-tit - always respond conflictually to
  cooperation and cooperatively towards conflict.

10
Game or Nash Equilibria

• Games also often have solutions or equilibrium
  points.
• These are outcomes which, owing to the selection
  of particular reasonable strategies will result
  in a determined outcome.
• An equilibrium is that point where it is not to
  either players advantage to unilaterally change
  his or her mind.

11
Saddle points

• The Nash equilibrium is also called a saddle
  point because of the two curves used to construct
  it
• an upward arching Maximin gain curve
• and a downward arc for minimum loss.
• Draw in 3-d, this has the general shape of a
  western saddle (or the shape of the universe and
  if you prefer). .

12
Some Simple Examples

• Battle of the Bismark Sea
• Prisoners Dilemma
• Chicken

13
The Battle of the Bismarck Sea

• Simple 2x2 Game
• US WWII Battle

Japanese Options Japanese Options
Sail North Sail South
US Options Recon North 2 Days 2 Days
US Options Recon South 1 Day 3 Days
14
The Battle of the Bismarck Sea
Japanese Options Japanese Options Japanese Options
Sail North Sail South Minima of Rows
US Options Recon North 2 Days 2 Days 2
US Options Recon South 1 Day 3 Days 1
Maxima of Columns Maxima of Columns 2 3
15
The Battle of the Bismarck Sea - examined

• This is an excellent example of a two-person
  zero-sum game with a Nash equilibrium point.
• Each side has reason to employ a particular
  strategy
• Maximin for US
• Minimax for Japanese).
• If both employ these strategies, then the outcome
  will be Sail North/Watch North.

16
Decision Tree
17
The Prisoners Dilemma

• The Prisoners dilemma is also 2-person game but
  not a zero-sum game.
• It also has an equilibrium point, and that is
  what makes it interesting.
• The Prisoner's dilemma is best interpreted via a
  story.

18
A Simple Prisoners Dilemma
Prisoner A Prisoner A
Confess Confess
Prisoner B Confess -1 -1 0 -10
Prisoner B Confess -10 0 -5 -5
19
Alternate Prisoners Dilemma Language
Uses Cooperate instead of Confess to denote
player cooperation with each other instead of
with prosecutor.
Prisoner A Prisoner A
Cooperate Defect
Prisoner B Cooperate -1 -1 0 -10
Prisoner B Defect -10 0 -5 -5
20
What Characterizes a Prisoners Dilemma
Uses Cooperate instead of Confess to denote
player cooperation with each other instead of
with prosecutor.
Prisoner A Prisoner A
Cooperate Defect
Prisoner B Cooperate Reward Reward Tempt Sucker
Prisoner B Defect Sucker Tempt Punish Punish
21
What makes a Game a Prisoners Dilemma?

• We can characterize the set of choices in a PD
  as
• Temptation (desire to double-cross other player)
• Reward (cooperate with other player)
• Punishment (play it safe)
• Sucker (the player who is double-crossed)
• A game is a Prisoners Dilemma whenever
• T gt R gt P gt S
• Or Temptation gt Reward gt Punishment gt Sucker

22
What is the Outcome of a PD?

• The saddle point is where both Confess
• This is the result of using a Minimax strategy.
• Two aspects of the game can make a difference.
• The game assumes no communication
• The strategies can be altered if there is
  sufficient trust between the players.

23
Solutions to PD?

• The Reward option is the joint optimal payoff.
• Can Prisoners reach this?
• Minimax strategies make this impossible
• Are there other strategies?

24
Iterated Play

• The PD is a single decision game in which the
  Nash equilibrium results from a dominant
  strategy.
• In iterated play (a series of PDs), conditional
  strategies can be selected

25
The Theory of Metagames

• Metagames step back from the game and look at the
  other players strategy
• Strategic choice is based upon opponents choice.
• For instance, we could adopt the following
  strategies
• Tit-for-tat
• Tat-for-tit
• Choose Confess regardless
• Choose Confess regardless

26
A Prisoners Dilemma Metagame
Prisoner A Prisoner A Prisoner A Prisoner A
Confess Regardless Confess Regardless Tit-for-tat Tat-for-tit
Pris B Confess -1 -1 0 -10 -1 -1 0 -10
Pris B Confess -10 0 -5 -5 -5 -5 -10 0
27
The Full PD Metagame

• See page 36 in Brahms
• Using the Metagame tit-for-tat strategy, we get
  three possible equilibria
• One the original both confess
• The other two, both confess (a cooperative
  solution)

28
The Morality of tit-for-tat

• Note that the conditional strategy of cooperate
  regardless is a direct manifestation of the
  Golden rule, perhaps the simplest and most
  ubiquitous moral maxim we have.
• Tit-for-tat invokes a rough system of justice (or
  equality in any event)
• Because tit-for-tat always results in equal
  payoff to both players, it is a very equitable
  strategy.
• It also teaches its morality, by encouraging
  mutual benefit through reciprocity.

29
Chicken

• The game that we call chicken is widely played in
  everyday life
• bicycles
• Cars
• Interpersonal relations
• And more

30
The Game of Chicken
Driver A Driver A
Swerve Swerve
Driver B Swerve 1 1 2 4
Driver B Swerve 4 2 3 3
31
Chicken is an Unstable game

• There is no saddle point in the game.
• No matter what the players choose, at least one
  player can unilaterally change for some
  advantage.
• Chicken is therefore unstable.
• We cannot predict the outcome

32
Chicken is Nuclear Deterrence
33
National Missile Defense

• Lets pick a current problem
• National Missile Defense
• Structure this as a game

34
The Game of National Missile Defense
US US
Build Build
Rogue State Attack 60B 5BG 1Tr 5BG
Rogue State Attack 60B 0 0 0
35
Calculating Expected Utility of NMD

• E(Build)pA(B-C)pA(B-C)
• E(Build)pA(B-C)pA(B-C)
• E(Build)pA(0-60)pA(0-60)
• E(Build)pA(0-1000)pA(0-0)
• Build NMD if E(Build)gtE(Build)

36
Spreadsheet

• Open Excel table

37
Utility Curves

• p1

38
Adding Complexity

• More players
• Information differences
• If players do not know the payoffs then they have
  very incomplete information
• Nested Games
• The

39
Tragedy of the Commons

• First observed during the British Enclosure
  movement
• Describes the problem of the unregulated use of a
  public good
• Take a commons e.g. a common pasture for
  grazing of cattle in a village

40
An example

• Take a village with 10 families
• Each family has 10 cows which just exactly
  provide the food they need.
• The village commons has a carrying capacity of
  100 cows

41
Cow Carrying Capacity

• Each cow produces 500 lbs of meat dairy per
  year up to or at carrying capacity of the
  pasture.
• 10 families X 10 Cows X 500 lbs 50,000
  lbs of food at carrying capacity
• and then Farmer Symthes wife has triplets

42
One more cow

• So Farmer Smythe decides he really needs one more
  cow.
• And there is no one to tell him no because the
  commons is an unregulated public good
• Like
• Air
• Water
• Security ?

43
Reduced Capacity

• With the overgrazing, each cow will now produce
  only 490 lbs of food.
• 10 families X 10 Cows X 490 lbs 49,000
  lbs of food at carrying capacity
• Each family gets 4900 lbs of meat dairy,
  instead of 5000.
• Except Farmer Smythe, who gets 5390 lbs
• Even with the reduced carrying capacity, it is
  still to his advantage to add the extra cow

44
Look Familiar?

• Look at the situation
• N players
• Equilibrium solution is to cooperate
• Joint optimal outcome is to cooperate,
• This is an n-person Prisoners dilemma

45
Thinking Strategically 10 Tales of Strategy

• Dixit and Nalebuff offer 10 simple concepts that
  are strategic in nature.
• They are worth reviewing, as they all demonstrate
  some particular strategic choice.

46
The Hot Hand

• Are hot hands just random sequences in a long
  series of trials?
• Probabilistic analysis suggests that what sports
  observers claim as periods of exceptional
  performance are not statistically excessive
• But hot hands may be masked by team responses.
• Take tennis
• If your backhand is weak, your opponent will play
  to it.
• Improve your backhand, and you get to use your
  better forehand more

47
To Lead or not to Lead

• Front-runner strategy
• The leading sailboat copies the strategy of the
  trailing boat.
• Doesnt work in 3 boat races
• Applicable to election?
• When to go negative?

48
Go Directly to Jail

• Just the Prisoners Dilemma

49
Here I stand

• Taking an irrevocable stand may change your
  opponents strategies.
• A public statement makes a commitment (and
  thereby changes payoffs) in ways that may dictate
  an outcome.
• Opponent has to take it or leave it.
• May be costly next time!

50
Belling the Cat

• Is the individual willing to assume the risks of
  the group
• This is a Hostages Dilemma
• Note the reference to plane full of passengers
  powerless before a hijacker with a gun.
• Is this likely to be the case after 9/11?
• What does this say about this strategic decision?

51
The Thin End of the Wedge

• Special interests get attention due to a
  one-at-time strategy
• We approve special provisions for a single
  request, when if we looked at the aggregate
  impact of many such requests, we might not
• Line item veto is strategy for dealing with it.

52
Look before you leap

• Look at long term consequences/commitments prior
  to decision
• E.g heroin

53
Mix your plays

• Rely on your best strategy strongest asset
• But not exclusively
• If you run the football every play, the defensive
  backfield will pull in and you will be less
  effective.
• The pass sets up the run.

54
Never give a sucker an even bet

• When someone offers to bet you, they often know
  the odds dont bet.
• Such as appliance warranties

55
Game theory can be dangerous to your health

• Check your bargaining position before you
  negotiate.
• Do you negotiate first or afterwards?
• Does your physical setting influence strategy?

56
Other Games

• Battle of the Sexes

57
Battle of the Sexes
M M
Football Ballet
F Football 2 1 0 0
F Ballet 0 0 1 2
58
Battle of the sexes