War of Attrition Models

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War of Attrition Models
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Games of Timing

• War of attrition models are one type of games of
  timing. These games are played in continuous
  time. Each player has only one decision when it
  will stop the game.
• Example Duel
• Strategies in games of timing specify when the
  player will stop the game, and what it will do
  when it stops the game.

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War of Attrition in Continuous Time

• In a war of attrition, the players compete for a
  prize. Call its value P. When one side quits,
  the other side gets the prize. Once the game
  starts, both sides accumulate costs as time
  passes.
• Let cA(t) cB(t) -t. Then if a player quits
  at time t, its payoff is -t, and the other
  players is P - t.

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Critical Stopping Times

• The time where a players accumulated costs equal
  the value of the prize is critical to solving
  wars of attrition.
• Example As critical time is P.
• A player will always quit by its critical time
  because it would prefer to quit at time 0 than
  continue the game after its critical time even if
  it wins the prize by doing so.

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Equilibrium with Complete Information

• In the symmetric case, we construct a symmetric
  mixed strategy equilibrium
• Let Q(t) be the cumulative quit rate for each
  player. Then optimal play means

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• Solution to this differential equation is
• Q(t) 1 - e-t/P
• The distribution of duration times declines over
  time.

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Asymmetric Costs

• Asymmetric costs is different. Let cA(t) -t and
  cB(t) -2t. Then if A quits at time t, As
  payoff is -t, and Bs is P - 2t.
• When both sides value for the prize and costs
  are common knowledge, the side with the higher
  cost quits immediately (t 0).
• Why? In the example, A knows that B will quit by
  time ½P. A can guarantee itself ½P by waiting
  until this time. Because this payoff is greater
  than As value for quitting any time before ½P, A
  will never quit before then.
• Consequently, B cannot win the prize and should
  quit at t 0.

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Fearons Audience Cost Model

• This model represents a crisis as a war of
  attrition, with the difference that only the side
  that quits pays the accumulated costs.
• Audience costs, whether domestic or
  international, only suffered by the loser of the
  crisis.
• Types given by value for war drawn from WA,0
  and WB,0 with cumulative distributions FA(w)
  and FB(w).

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• Each can quit the game by backing down or
  attacking. Backing down gives the other side the
  prize attacking gives both sides their war
  payoffs.
• If a side backs down, it suffers audience costs
  of -ai(t). For ease, let these costs accumulate
  linearly in time.

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• Lock-in is key strategic dynamic. A type is
  locked-in if -ai(t) lt wi as it would never back
  down.
• In equilibrium, there is a horizon. After the
  horizon, no type remaining quits it does not
  matter who attacks or when. Before the horizon,
  each sides probability of backing down is given
  by the following

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Bargaining Models vs. War of Attrition

• War of attrition models are another way to think
  about bargaining even though the parties cannot
  make offers other than the entire prize.
• Impressionist evidence from crises and wars about
  absence of give-and-take bargaining.
• Kennan and Wilson article on testing models of
  bargaining with data on strikes.