Lecture 3A Dominance

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Lecture 3ADominance

• This lecture shows how the strategic form can be
  used to solve games using the dominance principle.

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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.

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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.

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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.

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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.

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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.

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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.

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Katrina
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Strategies and payoff calculations for Katrina
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Strategic form of Katrina
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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
• Subjects are risk lovers, and like gamblers are
  willing to pay for the opportunity to gamble with
  nature.
• Subjects are confused by the calculations
  required to maximize expected value.

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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.

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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.

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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 .
  . .

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Rule 3

• Each player should discard his dominated
  strategies

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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.

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Strategies dominated by a mixture

• The supermarket's hours strategy is dominated by
  a mixture of the price and service strategies.
• Let p denote the probability that the supermarket
  chooses a price strategy, and (1-p) denote the
  probability that the supermarket chooses a
  service strategy.
• This mixture dominates the hours strategy if the
  following three conditions are satisfied
• p65(1-p)50 gt 45 or p gt -1/3
• p50(1-p)55 gt 52 or 3/5 gt p
• p60(1-p)50 gt 55 or p gt ½
• Hence all mixtures of p satisfying the
  inequalities
• ½ lt p lt 3/5
• dominate the hours strategy.

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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.
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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.

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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.
• Killingtons 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.

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Strategies in the ski resort game

• Killingtons strategies are to
• open
• close
• open only if it snows
• open only if it rains
• The MBAs strategies are to
• ski
• study

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Payoff calculations
For each strategy pair and corresponding matrix
cell, we compute the expected payoffs using the
probabilities of rain versus snow.
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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.

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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.

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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.

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Lecture 4AIterative Dominance

• This lecture continues our study of the
  strategic form, extending the principle of
  dominance to iterative dominance.

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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.

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Demand and Technology

• Consumer demand for a product is a linear
  function of price, and that market pre-testing
  has established
• We also suppose that the industry has constant
  scale returns, and we set the average cost of
  producing a unit at 1.

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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).

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Profit to the first firm

• As a function of (p1,p2), the net profit to the
  first firm is
• Net profit to the second firm is calculated in a
  similar way.

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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.

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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.

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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.

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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 q112 - q1 q2
• The profits of the second firm are calculated the
  same way.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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).

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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.

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Solving for Jarvis

• Since 4 gt 0, Jarvis bottles with cork if Mondavi
  does.
• The expected value of using screwtops when
  Mondavi does is
• (0.34 0.24 )/(0.2 0.3) 4.0
• while the expected value of retaining corking
  when Mondavi switches is
• (0.3 0.26)/(0.2 0.3) 3.0
• Therefore Jarvis always follows the lead of
  Mondavi.

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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 Mondavis 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?

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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 Mondavis superior market
  research increases Jarviss expected net profits
  by
• 0.22 0.4.
• This sets the upper bound Jarvis is willing to
  pay for independent research.

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Rule 4

• Rule 4 Iteratively eliminate
• dominated strategies.

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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.

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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.