Choices Involving Strategy

1
Chapter 12

• Choices Involving Strategy

2
Main Topics

• What is a game?
• Thinking strategically in one-stage games
• Nash equilibrium in one-stage games
• Games with multiple stages

12-2
3
What is a Game?

• A game is a situation in which
• each member of a group (or, each player) makes
  at least one decision, and
• Each players welfare depends on others choices
  as well as his own choice
• A game
• Includes any situation in which strategy plays a
  role
• Military planning, dating, auctions, negotiation,
  oligopoly

12-3
4
One-stage and multiple-stage games

• Two types of games
• One-stage game each player makes all choices
  before observing any choice by any other player
• Rock-Paper-Scissors, open-outcry auction
• Multiple-stage game at least one participant
  observes a choice by another participant before
  making some decision of her own
• Poker, Tic-Tac-Toe, sealed-bid auction

5
Figure 12.1 How to Describe a one-stage Game

• Essential features of a one-stage game
• Players
• Actions or strategies
• Payoffs
• Represented in a simple table
• The game is called
• Battle of Wits
• From The Princess Bride by S. Morgenstern
• Matching Pennies

This could be a metaphor for a battle in a war,
or for a tennis rally.
12-5
6
Thinking StrategicallyDominant Strategies

• Each player in the game knows that her payoff
  depends in part on what the other players do
• Needs to make a strategic decision, think about
  her own choice taking other players view into
  account
• A players best response is a strategy that
  yields her the highest payoff, assuming other
  players play specified strategies
• A strategy is a players dominant strategy if it
  is the players best response, no matter what
  strategies are chosen by other players

12-6
7
The Prisoners Dilemma Scenario

• Players Oskar and Roger, both students
• The situation they have been accused of cheating
  on an exam and are being questioned separately by
  a disciplinary committee
• Available strategies Squeal, Deny
• Payoffs
• If both deny, both suspended for 2 quarters
• If both squeal, both suspended for 5 quarters
• If one squeals while the other denies, the one
  who squeals is suspended for 1 quarter and the
  one who denies is suspended for 6 quarters

12-7
8
Figure 12.3 Best Responses to the Prisoners
Dilemma
(a) Oskars Best Response
(b) Rogers Best Response
Deny
Deny
Oskar
Oskar
Squeal
Squeal
12-8
9
Figure 12.4 Best Responses to the Provosts
Nephew
(a) Oskars Best Response
(b) Rogers Best Response
Deny
Deny
Oskar
Oskar
Squeal
Squeal
12-9
10
Thinking Strategically Iterative Deletion of
Dominated Strategies

• Even if the strategy to choose is not obvious,
  one can sometimes identify strategies a player
  will not choose
• A strategy is dominated if there is some other
  strategy that yields a strictly higher payoff
  regardless of others choices
• No sane player will select a dominated strategy
• Dominated strategies are irrelevant and can be
  removed from the game to form a simpler game
• Look again for dominated strategies, repeat until
  there are no dominated strategies left to remove
• Sometimes this allows us to solve games even when
  no player has a dominant strategy

12-10
11
Who will do what?
Top is dominated, for Al, by Low
Left is dominated, for Betty, by Right
High is dominated, for Al, by Bottom
Bottom is dominated, for Al, by Low
12
Guessing Half the Median

• This is another example of predicting the outcome
  of a game by means of the iterative deletion of
  dominated strategies
• Five players (actually, any odd number will do)
• Each picks a number up to a given maximum
• Each players penalty is the difference between
  his chosen number and half the median of the
  players chosen numbers
• Prove that each players choice is one (1)

13
Second-Price Sealed-Bid Auctions truth is the
weakly dominant strategy!

• The highest bidder wins the auction
• But pays the second-highest bid
• Why is this auction special?
• Every bidder has a weakly dominant strategy bid
  what the objects value is to you

14
Nash Equilibrium inOne-Stage Games

• Concept created by mathematician John Nash,
  published in 1950, awarded Nobel Prize
• Has become one of the most central and important
  concepts in microeconomics
• In a Nash equilibrium, the strategy played by
  each individual is a best response to the
  strategies played by everyone else
• Everyone correctly anticipates what everyone else
  will do and then chooses the best available
  alternative
• Combination of strategies in a Nash equilibrium
  is stable
• A Nash equilibrium is a self-enforcing agreement
  every party to it has an incentive to abide by
  it, assuming that others do the same

12-14
15
Figure 12.8 Nash Equilibrium in the Prisoners
Dilemma
12-15
16
Figure 12.9 The Battle of the Sexes
12-16
17
Nash Equilibria in Games with Finely Divisible
Choices

• Concept of Nash equilibrium also applies to
  strategic decisions that involve finely divisible
  quantities
• To find the Nash Equilibrium
• Determine each players best response function
• A best response function shows the relationship
  between one players choice and the others best
  response
• A pair of choices is a Nash equilibrium if it
  satisfies both response functions simultaneously

12-17
18
Figure 12.10 Free Riding in Groups
12-18
19
Mixed Strategies

• Can you find the Nash Equilibrium?
• There is none
• if only pure (or, non-random) strategies are
  allowed
• But if each player tosses a coin to pick his
  strategy, these randomized strategies are the
  Nash Equilibrium of this game.

12-19
20
Mixed Strategies

• When a player chooses a strategy without
  randomizing he is playing a pure strategy
• Some games have no Nash equilibrium in pure
  strategies. In these cases, look for equilibria
  in which players introduce randomness
• A player employs a mixed strategy when he uses a
  rule to randomize over the choice of a strategy
• Virtually all games have mixed strategy
  equilibria
• In a mixed strategy equilibrium, players choose
  mixed strategies and the strategy each chooses is
  a best response to the others players chosen
  strategies

12-20
21
Battle of Wits has a Nash Equilibrium in Mixed
Strategies
Similarly, given that Wesley is playing a mixed
strategy, Vizzini will also play a mixed strategy
only if his two pure strategies, Left and Right,
have equal payoffs. That is, p ? (-1) (1 - p) ?
1 p ? 1 (1 - p) ? (-1). This yields p 0.5.
Given that Vizzini is playing a mixed strategy,
Wesley will also play a mixed strategy only if
his two pure strategies, Left and Right, have
equal payoffs. That is, q ? 1 (1 - q) ? (-1)
q ? (-1) (1 - q) ? 1. This yields q 0.5.
In the Nash equilibrium of this game, both
players will play each strategy with a 50
probability.
22
Weakly-dominated strategies
There are two Nash equilibria (T, L) and (B, R).
But only the latter does not have a weakly
dominated strategy. Therefore, (B, R) is a better
prediction.
23
Games with Multiple Stages

• In most strategic settings events unfold over
  time
• Actions can provoke responses
• These are games with multiple stages
• In a game with perfect information, players make
  their choices one at a time and nothing is hidden
  from any player
• Multi-stage games of perfect information are
  described using tree diagrams

12-23
24
Figure 12.13 Lopsided Battle of the Sexes
12-24
25
Thinking StrategicallyBackward Induction

• To solve a game with perfect information
• Player should reason in reverse, start at the end
  of the tree diagram and work back to the
  beginning
• An early mover can figure out how a late mover
  will react, then identify his own best choice
• Backward induction is the process of solving a
  strategic problem by reasoning in reverse
• A strategy is one players plan for playing a
  game, for every situation that might come up
  during the course of play
• One can always find a Nash equilibrium in a
  multi-stage game of perfect information by using
  backward induction

12-25
26
A Two-Stage Game

• This game has two Nash equilibria
• Al chooses r and Betty chooses R, and
• Al chooses l and Betty chooses L.
• But only the former is subgame perfect
• In other words, only the former satisfies
  backward induction
• The (l, L) equilibrium is based on a non-credible
  threat by Betty to play L if the opportunity
  arose.

27
Cooperation in Repeated Games

• Cooperation can be sustained by the threat of
  punishment for bad behavior or the promise of
  reward for good behavior
• Threats and promises have to be credible
• A repeated game is formed by playing a simpler
  game many times in succession
• May be repeated a fixed number of times or
  indefinitely
• Repeated games allow players to reward or punish
  each other for past choices
• Repeated games can foster cooperation

12-27
28
Figure 12.16 The Spouses Dilemma

• Marge and Homer simultaneously choose whether to
  clean the house or loaf
• Both prefer loafing to cleaning, regardless of
  what the other chooses
• They are better off if both clean than if both
  loaf

12-28
29
Repeated Games Equilibrium Without Cooperation

• When a one-stage game is repeated, the
  equilibrium of the one-stage game is one Nash
  equilibrium of the repeated game
• Examples both players loafing in the Spouses
  dilemma, both players squealing in the Prisoners
  dilemma
• If either game is finitely repeated, the only
  Nash equilibrium is the same as the one-stage
  Nash equilibrium
• Any definite stopping point causes cooperation to
  unravel

12-29
30
Repeated Games Equilibria With Cooperation

• If the repeated game has no fixed stopping point,
  cooperation is possible
• One way to achieve this is through both players
  using grim strategies
• With grim strategies, the punishment for selfish
  behavior is permanent
• A credible threat of permanent punishment for
  non-cooperative behavior can be strong enough
  incentive to foster cooperation

12-30
31
The grim strategy may enforce cooperation
If Marge and Homer both play the grim strategy,
their payoffs are
If Marge plays the grim strategy but Homer
decides to loaf in Round 3, their payoffs are
Assuming Homer cares sufficiently about the
future losses that would occur if he decides to
loaf, it is a Nash equilibrium of the repeated
Spouses Dilemma when both Marge and Homer play
the grim strategy. Therefore, cooperation is
possible in indefinitely repeated games.
32
Asymmetric Information

• Now the true payoffs are not necessarily known to
  all players
• Each player knows his own payoffs but not
  necessarily the payoffs of the other players
• In these games, each players decision can reveal
  some of his information to the other players
• And each player can try to mislead the other
  players

33
Winners Curse

• A Ford Mustang is offered for sale by
  second-price auction
• There are three bidders Melissa, Olivia, and
  Elvis
• There is a 50 chance that the car has a serious
  mechanical problem, in which case it is worth
  2,000 to all bidders
• And there is a 50 chance that the car is
  problem-free, in which case it is worth 10,000
  to all bidders
• Melissa knows whether or not there is a problem
• Elvis and Olivia have no idea they are willing
  to pay 6,000, the expected value of the car
• What will happen at the auction?

34
Winners Curse

• If Elvis and Olivia do not know that Melissa
  knows the true condition of the car, the Nash
  equilibrium outcome is that Melissa bids the true
  value of the car and the others each bid 6,000
• Winners Curse If either Elvis or Olivia win,
  they have overpaid!
• If Elvis and Olivia know that Melissa knows the
  true condition of the car, the Nash equilibrium
  outcome is that Melissa bids the true value of
  the car and the others each bid 2,000
• No Winners Curse

35
Reputation