Dominance

1
Dominance

2
Overview

• In this unit, we explore the notion of dominant
  strategies
• Dominance often requires weaker views of
  rationality than does standard equilibrium play
• These weaker rationality requirements support
  choice of equilibria satisfying dominance over
  other equilibria.

3
An Example Prisoners Dilemma

• In this game, it pays to defect regardless of the
  rivals strategy
• Defect is a best response to cooperate
• Defect is a best response to defect

4
An Example Prisoners Dilemma

• In the language of dominance
• The cooperate strategy is strictly dominated by
  defect
• This means that the defect strategy gives
  strictly higher payoffs for Rowena than does
  cooperate

5
Rationality

• Rationality axiom 1 Never play a strictly
  dominated strategy regardless of your opponent
• Why?
• Even if you have serious doubts about the
  rationality of the other player.
• A dominated strategy does strictly worse than
  some other strategy
• regardless of your rivals play
• So it should be avoided.

6
Solving Using Dominance

• In the prisoners dilemma, we can solve the game
  purely by eliminating dominated strategies
• Since this elimination leaves each side only one
  undominated strategy, this pair constitutes an
  equilibrium.

7
Team Production

• Both the design and the production departments
  are required to produce some saleable output.
• The quality of the output determines the price
  for which it can be sold.
• For each unit of effort undertaken by either
  team, up to 10 units, profits increase by
  1.5million/unit. After that, it does not
  increase.

8
Costs of Effort

• It costs 1million per unit of effort in either
  department
• Effort is unobservable by management
• To compensate design and production, management
  has instituted a profit sharing plan whereby
  production and design each get one-third of the
  profits as compensation.

9
Optimal Effort

• From the perspective of the firm as a whole, each
  unit of effort up to 10 taken by design and
  production costs only 1m and has a return of 50
• Therefore from the firms perspective each
  department should exert 10 units of effort

10
Equilibrium Effort

• Notice that the design team needs to determine
  its level of effort not knowing the choice of the
  production team.
• What are its profits if design chooses effort e1
  and production chooses e2?
• Profit1 Profit share Cost of effort
• Profit1 (1/3)(1.5e1 1.5e2) e1

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Equilibrium Effort Continued

• Profit1 (1/3)(1.5e1 1.5e2) e1
• Notice that regardless of e2, Profit1 is
  decreasing in e1
• So any choice e1 0 is dominated by e1 0.
• Hence design exerts no special effort despite the
  profit sharing incentives
• The situation for production is analogous
• The conclusion is that both production and design
  will try to free ride off the efforts of the
  other and no effort will occur

12
Solving the Free Rider Problem

• Free rider problems appear in numerous settings
• Devising incentive schemes to solve these
  problems is critical
• What was wrong with the profit sharing scheme?

13
Bonuses

• Suppose that instead of doing a straight profit
  sharing arrangement, the firm uses a bonus system
  to compensate design and production.
• Recall that if production were efficient, profits
  would be 30m and the profit share gave away
  2/3rds of this amount or 20m.
• Instead, suppose that the firm pays each team a
  bonus of 10m 1 if they reach the profit
  target of 30m.

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Equilibrium Analysis

• Suppose that design expects production to work
  all-out to meet the target.
• To receive the bonus, design has to work all-out
  too.
• If it doesnt, then the analysis is as it was
  before but without even the profit sharing
  incentive---therefore design either works all-out
  or not at all.
• How do these situations compare?

15
Design Choices

• If design doesnt work, it earns zero
• If they works all-out, profits equal the bonus
  less the cost of effort, which nets design 1.
• Thus, it is better to work all-out than not at
  all, so a best response to productions working
  all-out is for design to do likewise
• Bottom line The structure of incentive schemes
  (as well as the total amount) can have a big
  effect on free-rider problems.

16
Iterative Elimination

• Recall that rationality axiom 1 prescribed that
  it was never a good idea to play a dominated
  strategy
• If you have some confidence of your rivals
  rationality, you might be willing to assume that
  she follows this axiom as well.
• This suggests that you should eliminate her
  dominated strategies in thinking about the game.

17
Quantity Setting Again

• Recall that in the quantity setting game from
  last class, it was never a good idea to choose
  output more than the monopoly output.
• Recall that the best response of firm 1 was
• q1 6 - .5q2
• So 1 should produce no more than 6 units ever.
• Likewise for 2.

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Graphically
q2
q1(q2)
qm
q2(q1)
q1
qM
19
Deleting Dominated Strategies
q2
q1(q2)
qm
q2(q1)
q1
qM
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New Best Responses
q2
q1(q2)
qm
q2(q1)
q1
qM
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Iteration

• If 1 knows that 2 will never choose more than 6
  units, then 1 should never choose less than 3
  units.
• So eliminating dominated strategies by 2, reduces
  the scope for sensible choices by 1 to q1 3,
  6.
• Same for 2.

22
Another Iteration

• 1s best response q1 6 - .5q2
• If 1 knows that q2 is between 3 and 6, the only
  outputs that make sense are between 3 and 4.5.

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Graphically
q2
qm
q1
qM
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Eliminate Dominated Strategies
q2
qm
q1
qM
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Conclusion

• If we continue this process, well end up with
  the Nash equilibrium.
• In games of strategic substitutes, the Nash
  equilibrium is also the dominance solvable
  outcome
• Notice that each stage of deletion corresponds to
  a level of rationality assumed on the part of
  your opponent

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Dominance Solvable Games

• To use dominance to solve a game
• Delete dominated strategies for each of the
  players
• Look at the smaller game with these strategies
  eliminated
• Now delete dominated strategies for each side
  from the smaller game
• Continue this process until no further deletion
  is possible
• If only single strategies remain, the game is
  dominance solvable

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More on Dominance Solutions

• Not all games are dominance solvable
• If after elimination, a small set of strategies
  remain for each player
• These strategies survive iterative dominance and
  are relatively more robust than others

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Weak Dominance

• To eliminate a strategy as being dominated, we
  required that some other strategy always be
  better no matter the rivals move
• Suppose we weaken this
• A strategy is weakly dominated if, no matter what
  the rival does, there is some strategy that does
  equally well and sometimes strictly better.

29
Voting Game

• Recall our voting game. All 3 voters agreed that
  voting the legislation down was better than
  passing it.
• Recall that a unanimous yes vote was an
  equilibrium.

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Voting Game - Dominance

• Can we use strict dominance to eliminate
  strategies?
• No.
• What about weak dominance?
• Notice that a yes vote by player 1 is weakly
  dominated by a no vote

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Voting Game - Dominance

• A yes vote by any player is weakly dominated by a
  no vote
• So one round of deletion of weakly dominated
  strategies gives us the unique equilibrium of No,
  No, No.

32
Auctions

• eBay and a number of other online auctions use
  proxy bidding rules
• Under a proxy bid, you enter a bid amount, but
  what you pay is determined by the second highest
  bid plus a small increment.
• Suppose that you know your willingness to pay for
  an item for sale on eBay.
• What should you bid?

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A Model of eBay

• Theres a lot of sniping on eBay
• Sniping is where bidders wait for the last
  possible instant to bid
• In that case, there is little feedback about
  other bids at the time you place your bid
• Think of the following version of the eBay game
• There are an unknown number of potential bidders
• You know your value, but know little about other
  bidders (including their rationality or their
  valuations)
• All bidders choose bids simultaneously
• High bid wins
• Pays second highest bid

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Bidders Problem

• How should you bid in this auction?
• It turns out that eliminating weakly dominated
  strategies provides an answer regardless of your
  rivals choice

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Graphically Bid Shading
Profit
If I shade down my bid, this is my profit profile
v
Highest rival bid
My bid
v
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Graphically Bidding Above Value
Profit
If I shade up my bid, this is my profit profile
v
My bid
Highest rival bid
v
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Graphically Bid Value
Profit
If bidvalue, this is my profit profile
v
Highest rival bid
v
My bid
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Comments

• Notice that when bid value
• I win in all the cases when bid
• And in some cases where I lost earlier.
• Moreover, these cases are profitable
• Notice that when bid value
• I win in fewer cases than when bid value
• But I made losses in all the cases where I won
  when bid value
• Therefore Im better off losing then

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Weak Dominance

• Therefore
• Bid value
• Does at least as well as all other strategies in
  many cases
• And strictly better in some cases
• So all other strategies are weakly dominated by
  bid value
• So we can use weak dominance (one round of
  deletion) to find the best strategy in this
  auction

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Case Study Tender Offers

• A frequent strategy among corporate raiders in
  the 80s was the two-tiered tender offer.
• Suppose the initial stock price is 100.
• In the event that a firm is taken private,
  shareholders get 90 per share.
• Campeau will buy shares a 105 for the first 50,
  and 90 for the remainder.

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Tenders...

• All shares are bought at the blended price of
  totals tendered.
• For instance, if z50 of shares are tendered,
  then the price is
• P 105 x (50/z) 90 x ((z-50)/z)
• P 90 15 x (50/z)

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Details

• Notice that the tender is a binding agreement to
  purchase shares regardless of the success of the
  takeover.
• Second, notice that if everyone tenders, the
  raider pays
• P 90 15 x (50/100) 97.50
• which is cheaper than the initial price of the
  stock!

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Dominance of the tender

• What is less obvious is that it is a dominant
  strategy to accept the tender
• Three cases to consider
• z50. Then P 90 15 x (50/z) 90
• z 100
• z50. Then P 105 100 or 90
• So it is a dominant strategy to sell your shares.

44
A White Knight

• Suppose Warren Buffet offers to buy all shares at
  102 conditional on getting a majority.
• Does this undo the two-tiered offer strategy?

45
Dominance revisited

• Again, consider the 3 cases
• z
• z 50. P 97.50 vs 90
• z 50. P 105 vs 100 or 102.
• Is there any way to undermine the two-tiered deal?

46
Summary

• Rationality Axiom Dont play dominated
  strategies
• As your confidence about the rationality of your
  opponent grows, can iteratively delete dominated
  strategies to arrive at a good plan
• Deletion of weakly dominated strategies can give
  clarity in even complicated situations