1. Choices Involving Strategy
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 A game
each member of a group (or, each “player”) makes at least one decision, and
Each player’s 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
Can you predict who will do what?
This could be a metaphor for a battle in a war, or for a tennis rally.
12-5

6. Thinking Strategically: Dominant 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 player’s dominant strategy if it is the player’s 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) Oskar’s Best Response
(b) Roger’s Best Response
Roger
Deny
Squeal -2 -1 -6 -5
Roger
Deny
Squeal -2 -1 -6 -5
Deny
Deny
Oskar
This game is a metaphor for all sorts of real-world situations in which the pursuit of self interest is a dominant strategy and leads to collective doom. Littering, pollution
Oskar
Squeal
Squeal
12-8

9. Figure 12.4: Best Responses to the Provost’s Nephew
(a) Oskar’s Best Response
(b) Roger’s Best Response
Roger
Deny
Squeal -2 -1 -6 -5
Roger
Deny
Squeal -2 -1 -6 -5
Deny
Deny
Oskar
Oskar is the Provost’s nephew. If both Oskar and Roger deny wrongdoing, the Provost can pull some strings on Oskar’s behalf. Roger still has a dominant strategy. That is enough to make (squeal, squeal) the obvious solution.
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? Left is dominated, for Betty, by Right
It may be possible to predict the outcome of a game by means of the iterative deletion of dominated strategies
Betty
Left
Center
Right Al
Top
3, 1
2, 3
10, 2
High
4, 5
3, 0
6, 4
Low
2, 2
5, 4
12, 3
Bottom
5, 6
9, 7
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 player’s penalty is the difference between his chosen number and half the median of the players’ chosen numbers
Prove that each player’s 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 object’s value is to you

14. Nash Equilibrium in One-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
Nash equilibria are “no regrets” equilibria: once everybody’s chosen strategies are revealed, nobody would wish they had chosen some other strategy.
12-14

15. Figure 12.8: Nash Equilibrium in the Prisoners’ Dilemma
Roger
Deny
Squeal
Oskar -2 -1 -6 -5
Dominant strategy equilibria are also Nash equilibria.
12-15

16. Figure 12.9: The Battle of the Sexes
Tony
Action
Romantic
Maria 5 2 -1 1
No dominant strategy equilibria. But there is a Nash equilibrium. In fact there are two! The BoS game is a metaphor for all sorts of situations that require coordination.
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 player’s best response function
A best response function shows the relationship between one player’s choice and the other’s 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
Vizzini
Left
Right
Mixed (q, 1-q)
Wesley
1, -1
-1, 1
q  1 + (1 - q)  (-1), q  (-1) + (1 - q)  1
q  (-1) + (1 - q)  1, q  1 + (1 - q)  (-1)
Mixed (p, 1-p)
p  1 + (1 - p)  (-1), p  (-1) + (1 - p)  1
p  (-1) + (1 - p)  1, p  1 + (1 - p)  (-1)
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.
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.
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.
Betty L R Al T
6, 10
8, 10 B
5, 1
11, 5

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
Tony benefits from being the first mover. He can play the role of coordinator. So, he can force the equilibrium that’s favorable to him.
12-24

25. Thinking Strategically: Backward 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 player’s 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. l r L R
Al’s move
Betty’s move
(8, 10)
(5, 1)
(11, 5)
Nash Equilibrium, but not Subgame Perfect
Subgame Perfect Nash Equilibrium
Informally discuss equilibrium selection and Reinhard Selten ( ).

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:
Round 1
Round 2 3 4 5 6 …
Marge 2
Homer
If Marge plays the grim strategy but Homer decides to loaf in Round 3, their payoffs are:
Round 1
Round 2 3 4 5 6 …
Marge 2 1
Homer
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 player’s decision can reveal some of his information to the other players
And each player can try to mislead the other players

33. Winner’s 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. Winner’s 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
Winner’s 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 Winner’s Curse

35. Reputation