Lecture 1B Dominance

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Lecture 1B Dominance

• This lecture shows how the strategic form can be
  used to derive the solution if each player has a
  dominant strategy. Iterative dominance provides a
  further for way of restricting the number of
  strategies that might be desirable. It is based
  on the idea that every player in the game
  believes that no one will play a dominated
  strategy.
• Read Chapters 7 and 8 of Strategic Play.

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Games with imperfect information

• In many situations, you must determine your
  strategy without knowing what your rival is
  doing, and his situation is similar to yours.
• Even if the moves are not literally taking place
  at the same moment, but both moves are made in
  ignorance of the rivals, the moves are
  effectively simultaneous.

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Simultaneous move games

• A game where no player can make a choice that
  depends on the moves of the other players is
  called a simultaneous move game.
• The strategic form of simultaneous move games has
  special significance, because in contrast to all
  other games, no information is lost when
  transforming a simultaneous move games from its
  extensive form to its strategic form.

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Acquiring Federated Department Stores

• Robert Campeau and Macy's are competing for
  control of Federated Department Stores in 1988.
• If both offers fail, then the market price will
  be benchmarked at 100. If one succeeds, then any
  shares not tendered to the winner will be bought
  from the current owner for 90.
• The argument here is that losing minority
  shareholders will get burned by the new majority
  shareholders.

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Campeaus offer . . .

• Campeau made an unconditional two tier offer. The
  price paid per share would depend on what
  fraction of the company Campeau was offered.
• If Campeau got less than half, it would pay 105
  per share. If it got more than half, it would pay
  105 on the first half of the company, and 90 on
  any remaining shares.
• Each share tendered would receive a blend of
  these two prices so that every share received the
  average price paid. If a percentage x gt 50 of the
  company is tendered, then 50/x of them get 105,
  and (1 - 50/x) of them get 90 for a blended price
  of
• 105 50/x 90(1 - 50/x) 90 15(50/x).

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Macys offer . . .

• Macy's offer was conditional at a price of 102
  per share it offered to pay 102 for each share
  tendered, but only if at least 50 of the shares
  were tendered to it.
• Note that if everyone tenders to Macy's, they
  receive 102 per share, while if everyone tenders
  to Campeau, they receive 97.50. so, shareholders
  are collectively better off tendering to Macy's
  than to Campeau.

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The payoff matrix to a stockholder
Although share holders are better off as a group
tendering to Macys, each individual shareholder
is better off tendering to Campeau, because it is
a dominant strategy.
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Strictly dominant strategies

• Strategies that are optimal for a player
  regardless of whether the other players play
  rationally or not are called dominant.
• If a dominant strategy is unique, it is called
  strictly dominant.
• Although a player's payoff might depend on the
  choices of the other players, when a dominant
  strategy exists, the player has no reason to
  introspect about the objectives of the other
  players in order to make his own decision.

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Rule 2

• If you have a dominant strategy, use it.

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A second way of representing games

• Rather than describe a game by its extensive
  form, one can describe its strategic form.
• The strategic form of the game is a list of all
  the possible pure strategies for each of the
  players and the (expected) payoffs resulting from
  them.
• Suppose every player chooses a pure strategy, and
  that nature does not play any role in the game.
  In that case, the strategy profile would yield a
  unique terminal node and thus map into payoffs.

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Strategies

• The foundation of the strategic form is a
  strategy.
• A strategy is a full set of instructions to a
  player, telling her how to move at all the
  decision nodes assigned to her.
• Strategies respect information sets the set of
  possible instructions at decision nodes belonging
  to the same information set must be identical.
• Strategies are exhaustive they include
  directions about moves the player should make
  should she reach any of her assigned nodes. The
  set of a players strategies is called the
  strategy space.

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Strategic form

• The strategic form representation is less
  comprehensive than the extensive form, discarding
  detail about the order in which moves are taken.
• The strategic form defines a game by the set of
  strategies available to all the players and the
  payoffs induced by them.
• In two player games, a matrix shows the payoffs
  as a mapping of the strategies of each player.
  Each row (column) of the table corresponds to a
  pure strategy. The cells of the table
  respectively depict the payoffs for the row and
  column player.

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Imperfect monitoring

• If the actions of those paid to make decisions on
  your behalf are hidden from you, whether you
  retain them or not cannot directly depend on what
  they do.
• This creates a situation of moral hazard, because
  you cannot directly reward them for following
  your instructions.
• In that case their employment contract with you
  typically depends on signals that are correlated
  with their behavior.

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Investment broker

• Mr. Madoff could be sinking his money into real
  stocks, or meeting debts incurred with former
  clients and running a Ponzi scheme.
• The client, knows more than the first mover, her
  broker, because she has the opportunity to revise
  her portfolio after reviewing data on the
  economy.
• For that reason, this game is neither a
  simultaneous move game, nor a perfect information
  game.

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Strategic form of investment broker

• The easiest way of solving this game is to
  directly analyze its strategic form.
• The strategies for each player are shown in the
  matrix.
• To obtain the payoffs suppose, for example, the
  broker chooses tech and the clients strategy
  is continue with broker. Then the brokers
  expected compensation is
• 0.53 0.59 6

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Solution to investment broker

• This game is dominance solvable.
• The broker should choose tech because it is a
  dominant strategy.
• The investor should use the signal she receives
  about the economy, picking the strategy continue
  if new, liquidate if bubble.

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MBA market

• CMU, Pitt and Duquesne compete in their MBA
  evening programs, drawing from an overlapping
  demand pool.
• Their reputations and cross synergies with other
  programs effectively shape the kinds of choices
  they offer.
• One of the players has a dominant strategy, and
  the game can be solved using iterative dominance.

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Marketing groceries

• In this simultaneous move 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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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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Rule 3

• All players should iteratively discard
  dominated strategies.

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Lecture summary

• Some games are easier to analyze in their
  strategic form than in their extensive form.
• The two most important principles for strategic
  play, are to play dominant strategies, and
  iteratively eliminate dominated strategies.
• The first principle applies regardless of whether
  the other players are rational or not, and
  therefore does not depend on whether you know
  their payoffs or not.
• The second principle applies when you know enough
  about the payoffs of the other players to
  recognize their dominated strategies.