War of Attrition

1
War of Attrition

• Delay under complete information

2
Set up

• Two sides are engaged in a costly conflict
• As long as neither side concedes, it costs each
  side 1 per period
• Once one side concedes, the other wins a prize
  worth Vgt1.
• V is a common value and is commonly known by both
  parties
• If both concede at the same time they split V
  evenly.
• What advice can you give for this game?

3
Pure Strategy Equilibria

• Suppose that player 1 will concede after t1
  periods and player 2 after t2 periods
• Where 0 lt t1 lt t2
• Is this an equilibrium?
• No 1 should concede immediately in that case
• This is true of any equilibrium of this type

4
More Pure Strategy Equilibria

• Suppose 1 concedes immediately
• Suppose 2 never concedes
• This is an equilibrium though 2s strategy is not
  credible
• The equilibrium must call for 1 conceding in
  period 0 and 2 conceding in period 1.
• But then, if Vgt2 player 1 should wait till the
  first period and grab V by paying 2.

5
Symmetric Pure Strategy Equilibria

• Suppose 1 and 2 will concede at time t.
• Is this an equilibrium?
• If Vgt2, then by not conceding each part gets more
  than 1. So by waiting you can grab more than one
  for you and pay the cost of 1.
• If Vlt2, then each part grabs less than 1 and by
  conceding earlier you can save the cost of 1.
• If V2, and
• t0 waiting is better so this is not an
  equilibrium.
• t1, it is an equilibrium.
• tgt1 this is not an equilibrium since by conceding
  at the beginning you save a on waiting costs.

6
Symmetric Equilibrium

• There is a symmetric equilibrium in this game,
  but it is in mixed strategies
• Suppose each party concedes with probability p in
  each period
• For this to be an equilibrium, it must leave the
  other side indifferent between conceding and not

7
When to concede

• Suppose up to time t, no one has conceded
• If I concede now, my payoff is t
• If I wait another period, my payoff is
• V (t1) if my rival concedes
• (t 1) if not
• Notice the t term is irrelevant
• Indifference
• -t (V (t1)) p ( (t 1)) (1-p)
• -t V p (t 1)
• 0 V p 1
• 1 / V p

8
Observations

• The mixed strategy equilibrium says that the
  probability of a concession for each player is
  constant and equal to 1/ V.
• The larger it is V the less probable somebody
  will concede.
• The larger are the stakes (V), the longer the
  expected duration of the war.
• Conditional on the war lasting until time t, the
  future expected duration of the war is exactly as
  long as it was when the war started
• In principle there can be no concessions in this
  game the negotiation takes forever.

9
Observations

• Since the equilibrium is in mixed strategies,
  each player is indifferent between continuing and
  stopping.
• The larger it is V the less probable somebody
  will concede.
• The larger are the stakes (V), the longer the
  expected duration of the war.
• Conditional on the war lasting until time t, the
  future expected duration of the war is exactly as
  long as it was when the war started
• In principle there can be no concessions in this
  game the negotiation takes forever.

10
Expected cost I
11
Expected cost II

• The total expected benefit is just that benefit
  multiplied by 2 TB (3V-2) / (2V-1)
• It is easy to show that TB lt V for Vgt1
• There are no economic profits to be had in a war
  of attrition with a symmetric rival.
• Look for the warning signs of wars of attrition

12
Wars of Attrition in Practice

• Patent races
• RD races
• Browser wars
• Costly negotiations
• Brinkmanship

13
War of Attrition II

• Asymmetric players

14
Set up

• Two stores in the same block are suffering the
  effects of a new supermarket in the neighborhood
• If both stores are open they lose 1000 per
  month.
• If only one remains open the present discounted
  value of the store earnings is 5000
• Once closed they can not reopen
• What is the symmetric equilibrium in mixed
  strategies?

15
Equilibrium

• Let pi be the probability that store i 1,2
  closes
• So store 1 gets
• For staying open p2 5000 (1- p2) 1000
• For closing 0
• Store 1 is indifferent if store 2 uses p2 1 / 6
• Store 2 is indifferent if store 1 uses p1 1 / 6

16
Set up II

• Now imagine that store 1 is a little bigger
• If both stores are open
• store 1 loses 1200 per month.
• store 2 loses 1000 per month.
• If only one remains open the present discounted
  value of the store earnings is
• 5000 for store 2
• V gt 5000 for store 1
• What is the symmetric equilibrium in mixed
  strategies?

17
Equilibrium II

• Let pi be the probability that store i 1,2
  closes
• So store 1 gets
• For staying open p2 V (1- p2) 1200
• For closing 0
• Store 1 is indifferent if store 2 uses p2 1200
  / (V 1200)
• Store 2 is indifferent if store 1 uses p1 1 / 6

18
Who is more likely to close
p1 , p2
6 / 31
p1 (V)
1 / 6
p2 (V)
V
6000
5000
19
Who is more likely to close II
p1 , p2
Store 2 is more likely to close
Store 1 is more likely to close
6 / 31
p1 (V)
1 / 6
p2 (V)
V
6000
5000