Lecture 3A
Dominance
This lecture shows how the strategic
form can be used to solve games
using the dominance principle.
Auctions
Auctions are widely used by companies, private
individuals and government agencies to buy and
sell commodities.
They are also used in competitive contracting
between an (auctioneer) firm and other (bidder)
firms up or down the supply chain to reach
trading agreements.
Here we compare two sealed bid auctions,
where there are two bidders with known
valuations of 2 and 4 respectively.
First price sealed bid auction
In a first price sealed auction,
players simultaneously submit
their bids, the highest bidder
wins the auction, and pays
what she bid for the item.
Highway contracts typically
follow this form.
For example, if the valuation 4
player bids 5 and the valuation
2 player bids 1, then the former
wins the auction, pays 5 and
makes a net loss of 1.
Dominance in a first price auction
There is a weakly
dominant strategy
for the row player
to bid 1.
Eliminating all the
other rows then
leads the column
player to maximize
his net earnings by
bidding 2.
Second price sealed bid auction
In a second priced sealed bid
auction, players simultaneously
submit their bids, the highest
bidder wins the auction, and
pays the second highest bid.
This is similar to Ebay, although
Ebay is not sealed bid.
For example, if the valuation 4
player bids 5 and the valuation 2
player bids 1, then the former
wins the auction, pays 1 and
makes a net profit of 3.
Dominance
in a second price auction
The row player has a
dominant strategy of
bidding 2.
The column player has
a dominant strategy of
bidding 4.
Thus both players have
a dominant strategy of
bidding their (known)
valuation.
Comparing two auction mechanisms
Comparing the two auctions, the bidder place
different bids but the outcome is the same,
the column player paying 2 for the item.
How robust is this result, that the form of the
auction does not really matter?
The first part of the strategy course sequel
45-975, Auction and Market Strategy, analyzes
this question in depth, and more generally,
investigates optimal bidding and auction
design.
Katrina
Strategies and payoff calculations
for Katrina
Strategic form of Katrina
Why might outcomes depend on the
way games are presented?
The difference between the outcomes in the
strategic form versus the extensive forms is
remarkable.
There are reasons why subjects “stay open” in the
extensive form and “close shop” in the strategic
form:
1. Subjects are risk lovers, and like gamblers
are willing to pay for the opportunity to
gamble with nature.
2. Subjects are confused by the calculations
required to maximize expected value.
The field manager
If a field manager can fool his supervisor, then
fabrication maximizes his compensation and career
prospects. But his worst outcome is to get caught.
Also the field manager is given more credit from the
firm if his supervisor conducts a diligent review of a
sound business proposal, than if his supervisor does
not thoroughly review it.
The regional supervisor
The regional supervisor is rewarded by the firm when
he detects problems in the field, and also when his
field manager makes sound business proposals.
If the supervisor is not diligent he cannot detect self
serving behavior, but he can recognize sloth.
Also diligent checking interferes with his other
activities at work and home.
Supervision
Games with dominated strategies do not necessarily
have dominant strategies.
Here we see to “Propose a least effort alternative” is
dominated by “Diligently create . . .”
Rule 3
Each player should discard his
dominated strategies
Marketing groceries
In this game the
corner store franchise
would suffer greatly if
it competed on the
same feature as the
supermarket.
This is illustrated by
the fact that its
smallest payoffs lie
down the diagonal.
Strategies dominated by a mixture
The supermarket's hours strategy is dominated by a
mixture of the price and service strategies.
Let π denote the probability that the supermarket chooses
a price strategy, and (1-π) denote the probability that the
supermarket chooses a service strategy.
This mixture dominates the hours strategy if the following
three conditions are satisfied:
π65+(1-π)50 > 45
or
π > -1/3
π50+(1-π)55 > 52
or
3/5 > π
π60+(1-π)50 > 55
or
π>½
Hence all mixtures of π satisfying the inequalities:
½ < π < 3/5
dominate the hours strategy.
Eliminating a dominated strategy
Upon eliminating
the hours strategy
from the game,
we see that a
dominant strategy
for the corner
store emerges,
that is choosing
“hours”.
Killington
In this game, an MBA
student can either
study on the weekend
or, if the resort is
open, ski.
The resort, Killington,
may decide to stay
open even if rain
turns the slopes to
mud and ice.
Rationalizing the payoffs
for the ski resort game
The MBA student prefers to skiing to study if it snows,
but prefers study to skiing if it rains.
Given her preference ordering, one can prove that the
solution of the game is not affected by the values the
MBA student places on each outcome.
Killington’s profits whenever the MBA student skis, but
makes higher profits if it snows.
Killington makes losses if they open and the student
studies. Those losses are smaller if it snows, because
its employees have the slopes to themselves.
Strategies in the ski resort game
Killington’s strategies are to:
1. open
2. close
3. open only if it snows
4. open only if it rains
The MBA’s strategies are to:
1. ski
2. study
Payoff calculations
For each strategy
pair and
corresponding
matrix cell, we
compute the
expected payoffs
using the
probabilities of
rain versus snow.
Strategic form of ski resort game
The strategic
form for this
game is easier to
analyze than its
extensive form.
The bottom
strategy of
Killington is
dominated by the
one above it.
Iterating further
If the MBA believes Killington does not play
dominated strategies, then he would eliminate the
bottom strategy from consideration, revealing a
dominant strategy to “ski”.
If Killington knows the MBA will reason in this
fashion, then its best response is to stay open
regardless of the whether.
Lecture summary
Some games are easier to analyze when
presented in their strategic form than in
extensive form.
We derived a third rule that applies to the
strategic form: do not play dominated strategies.
Our experiments also suggest that we might
extend the dominance principle. If a player
recognizes that another player will apply Rules 2
and 3, this may simplify the game for her.
Lecture 4A
Iterative Dominance
This lecture continues our study of the
strategic form, extending the principle of
dominance to iterative dominance.
Market games
Our next pair of examples illustrate how
the strategy space can greatly affect the
profitability of firms competing in a
concentrated industry.
Suppose there are just two firms in the
industry. We shall see that their market
value depends on whether they compete
on price, or on quantity.
Demand and Technology
Consumer demand for a product is a linear function
of price, and that market pre-testing has
established:
q  p 13 p
We also suppose that the industry has constant scale
returns, and we set the average cost of producing a
unit at 1.
Price competition
When firms compete on price, the firm which
charges the lowest price captures all the market.
When both firms charge the same price, they share
the market equally.
These sharp predictions would be weakened if
there were capacity constraints, or if there was
some product differentiation (such as location rents
or market niches).
Profit to the first firm
As a function of (p1,p2), the net profit to the first firm
is:
1 ( p1 , p2 )  (  p1 )( p1  c) if p1  p2
1
 (  p1 )( p1  c ) if
2
0 if
p1  p2
p2  p1
Net profit to the second firm is calculated in a similar
way.
Market games with price competition
In our example q = 13 – p
and c= 1.
We could try to solve the
problem algebraically.
An alternative is to see
how human subjects attack
this problem within an
experiment.
We have substituted some
price pairs and their
corresponding profits into
the depicted matrix.
Solving the price setting game
Setting price equals 7 is
dominated by a mixture of
setting price to 5 or 2, with most
of the probability on 5.
Eliminating price equals 7 for
both firms we are left with a 3 by
3 matrix.
Now setting price equals 5 is
dominated by a mixture of
setting price to 3 or 2.
In the resulting 2 by 2 matrix a
dominant strategy of charging 2
emerges for both players.
Quantity competition
When firms compete on quantity, demanders set a
market price that clears inventories and fills every
customer order.
If firms have the same constant costs of
production, and hence the same markup, then their
profits are proportional to their market share.
This predictions might be violated if the price
setting mechanism was not efficient, or if the
assumptions about costs were invalid.
Calculating profits when there is
quantity competition
Letting q1 and q2 denote the quantities chosen by the
firms, the industry price is derived from the demand
curve as:
p = ( - q1 – q2)/ = 13 - q1 – q2
When the second firm produces q2, as a function of
its choice q1, the profits to the first firm are
q1( - q1 – q2)/ - q1c = q1[12 - q1 – q2]
The profits of the second firm are calculated the
same way.
Market games with quantity competition
As in the price setting
game, we could try to solve
the game algebraically, or
set the model up as an
experiment.
If we can compute profits
as a function of the
quantity choices, using the
second approach, we can
easily vary the underlying
assumptions to investigate
the outcomes of alternative
formulations.
Solving the the quantity setting game
For both firms, setting
quantity equals 6 is
dominated by setting
quantity equals 5
Eliminating the strategy
of choosing 6 for both
firms, we are left with a
3 by 3 matrix in which
the weakly dominant
strategy is to pick
quantity equals 4.
Iterative dominance
Rules 2 and 3 rely on a player recognizing strategies
to play or avoid independently of how others behave.
If all players recognized situations in which these two
rules applied and abided by them, and one of the
players realized that, then this particular player should
exploit this knowledge to his own advantage by
refining the set of strategies the other players will use.
Knowing which strategies the other players have
eliminated reduced the dimension of his problem,
ruling out possible courses of action that might
otherwise look reasonable.
Is the algorithm of iteratively removing
dominated strategies unique?
Question: Can we have different solutions if we
use different sequence of truncations?
Answer: No
Fact: Different algorithms for eliminating strictly
dominated strategies lead to the same set of
solutions.
The key to proving this point is that if a strategy
is revealed to be dominated it will remain
dominated if another strategy is removed first.
How sophisticated are the players?
Applying the principle of iterative dominance assumes
players are more sophisticated than applying the
principle of dominance.
Applying the dominance principle in simultaneous move
games makes sense as a unilateral strategy.
In contrast, a player who follows the principle of iterative
dominance does so because he believes the other
players choose according to that principle too.
Each player must recognize all the dominated strategies
of every player, reduce the strategy space of every player
as called for, and then repeat the process.
Bottling wine
Corks are traditionally used in bottling wine,
but recent research shows that screwtops
give a better seal, and hence the reduce the
risk of oxidation and tainting. They are also
less expensive.
However consumers associate screwtops
with cheaper varieties of wine, so wineries
risk losing brand reputation from moving too
quickly ahead of the consumer tastes.
To illustrate this problem consider two Napa
valley wineries who face the choice of
immediately introducing screwtops or
delaying their introduction.
Extensive form game
Mondavi has
resources to
conduct market
research into
this issue, but
Jarvis does not.
However Jarvis
can retool more
quickly than its
larger rival, so it
can copy what
Mondavi does.
Strategies for Mondavi
A strategy for Mondavi is
whether to introduce
screwtops, abbreviated a
“y”, or retain corking,
abbreviated by “n”, for each
possible triplet of consumer
preferences.
Therefore Mondavi has 8
different strategies.
Reviewing the payoffs in
the extensive form, the
unique dominant strategy
for Mondavi is (n,y,y).
cork screwtop indifferent
y
y
y
y
y
n
y
n
y
n
y
y
y
n
n
n
y
n
n
n
y
n
n
n
Eliminating the
dominated strategies of Mondavi
We can simplify the
problem that Jarvis
has by drawing its
decision problem
when Mondavi
follows its dominant
strategy.
Solving for Jarvis
Since 4 > 0, Jarvis bottles with cork if Mondavi does.
The expected value of using screwtops when
Mondavi does is:
(0.3*4 + 0.2*4 )/(0.2 +0.3) = 4.0
while the expected value of retaining corking when
Mondavi switches is:
(0.3 + 0.2*6)/(0.2 +0.3) = 3.0
Therefore Jarvis always follows the lead of Mondavi.
Rivals as a source of information
The solution to this game shows that rivals can be a
valuable source of information.
Although Jarvis could undertake its own research into
bottling, it eliminates these costs by piggybacking off
Mondavi’s extensive marketing research.
Nevertheless Jarvis receives a noisy signal from
Mondavi. Jarvis cannot tell whether consumers prefer
screwtops or are indifferent.
How much would Jarvis be prepared to pay to conduct
its own research, and receive a clear signal?
The value of independent research
When consumers are
indifferent Jarvis could
capture a niche market by
corking, increasing its
profits by 6 – 4 = 2.
Hence access to Mondavi’s
superior market research
increases Jarvis’s expected
net profits by:
0.2*2 = 0.4.
This sets the upper bound
Jarvis is willing to pay for
independent research.
Rule 4
Rule 4: Iteratively eliminate
dominated strategies.
Four rules for good strategic play
Rule 1: Look ahead and reason back
Rule 2: If there is a dominant strategy, play it
Rule 3: Discard dominated strategies.
Rule 4: Iteratively eliminate dominated strategies.
Lecture summary
The second two rules, “play dominant
strategies” and “do not play dominated
strategies”, apply independently of whether
the other players are rational or not.
In this lecture we advocated using a fourth
rule that applies to the strategic form:
“iteratively eliminate dominated strategies”.
Like our first rule, “look forward and reason
back”, the fourth rule assumes that the other
players are rational. In this case we are
assuming that they will also apply the fourth
rule for their own purposes.