# Dominance - PowerPoint

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Dominance
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.
An Example – Prisoner’s Dilemma
Cooperate   Defect    In this game, it pays to
defect regardless of the
rival’s strategy
 Defect is a best
Cooperate   3, 3        0, 4       response to cooperate
 Defect is a best
response to defect
Defect      4, 0        1, 1
An Example – Prisoner’s Dilemma
Cooperate   Defect    In the language of
dominance:
   The cooperate strategy
is strictly dominated by
Cooperate   3, 3        0, 4            defect
   This means that the
defect strategy gives
strictly higher payoffs
Defect                                  for Rowena than does
4, 0        1, 1
cooperate
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 rival’s play
   So it should be avoided.
Solving Using Dominance
Cooperate   Defect    In the prisoner’s
dilemma, we can solve
the game purely by
eliminating dominated
Cooperate   3, 3        0, 4       strategies
 Since this elimination
leaves each side only
one undominated
Defect      4, 0        1, 1       strategy, this pair
constitutes an
equilibrium.
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.
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.
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 firm’s perspective each
department should exert 10 units of effort
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
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
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?
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.
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 doesn’t, 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?
Design Choices
 If design doesn’t 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 production’s 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.
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 rival’s
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.
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
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
Weak Dominance
 To eliminate a strategy as being dominated,
we required that some other strategy always
be better no matter the rival’s 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.
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?
A Model of eBay
 There’s 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
bidders (including their rationality or their valuations)
 All bidders choose bids simultaneously
 High bid wins

 Pays second highest bid
Bidder’s Problem
 How should you bid in this auction?
 It turns out that eliminating weakly dominated
strategies provides an answer regardless of

Profit       If I shade down my bid, this is my profit profile

v

My bid                          Highest rival bid
v
Graphically – Bidding Above Value

Profit       If I shade up my bid, this is my profit profile

v

My bid

Highest rival bid
v
Graphically – Bid = Value

Profit       If bid=value, this is my profit profile

v

Highest rival bid
v
My bid
 Notice that when bid = value
   I win in all the cases when bid < value
   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 I’m better off losing then
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
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.
Tenders...
 All shares are bought at the blended price of
totals tendered.
   For instance, if z%>50% of shares are
tendered, then the price is
   P =\$105 x (50/z) + \$90 x ((z-50)/z)
   P = \$90 + \$15 x (50/z)
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!
Dominance of the tender
 What is less obvious is that it is a dominant
strategy to accept the tender:
   Three cases to consider:
   z>50. Then P = \$90 + \$15 x (50/z) > 90
   z<50. Then P = \$105 > 100
   z=50. Then P = \$105 > 100 or 90
   So it is a dominant strategy to sell your
shares.
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?
Dominance revisited
 Again, consider the 3 cases:
   z < 50. P = \$105 vs \$100 or \$102.
   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?
Summary
 Rationality Axiom: Don’t play dominated
strategies