Mixed Strategy Nash Equilibrium in War of Attrition Game

Key Concepts

• Mixed Strategy Nash Equilibrium (MSNE)
  • Different forms of MSNE require different calculations.
  • Focus on stationary mixed strategy Nash equilibrium.

Stationary Mixed Strategy Nash Equilibrium

• Definition: At every period, player i drops out with probability ( p_i ) and waits with probability ( 1 - p_i ).
• Probabilities are constant across periods (e.g., period 0, 10, 1000).

Players' Perspectives

• Expected utility for player 1:
  • Notation: ( \sigma_1, \sigma_2^* ) are strategies based on probabilities of drop/wait for players 1 and 2.
  • Need to ensure players are indifferent between dropping and waiting.*

Expected Payoff Calculation

1. Dropping in Period t:
   • Player 1 loses and incurs a cost ( l_1(t) ).
2. Waiting in Period t, then Dropping in t+1:
   • Game may end in period t if opponent drops with probability ( p_2 ).
   • Expected payoff:
     • If opponent drops: ( h_1(t) )
     • If opponent waits: incurs cost ( l_1(t+1) )

Equivalence of Payoffs

• Drop at time t must yield the same expected utility as waiting and dropping at t+1:
  • ( l_1(t) = p_2 h_1(t) + (1 - p_2) l_1(t + 1) )

Solving for Equilibrium Probabilities

• Rearrange the equation to isolate probabilities.
• Conclude that:
  • ( p_2 = \frac{c_1}{v_1 + c_1} )
  • Symmetrical approach yields ( p_1 = \frac{c_2}{v_2 + c_2} )

Observations

• When costs are small, probabilities ( p_1^* ) and ( p_2^* ) approach 0.
• High likelihood of continuing the game for a long time, but eventual outcomes tend toward zero payoff.
• Expected payoff for both players (before gameplay) is 0.
  • Indifference: Each player is indifferent between dropping now or later, leading to equivalence.

Comparison with Pure Strategies

• In a pure strategy:
  • One player gets zero while the other gets their respective value.
• Mixed strategies lead to both players receiving zero, which is inefficient.
• However, mixed strategies exhibit properties like evolutionary stability.

Real-Life Implications

• Mixed strategies can be applicable in real-world coordination problems, similar to the war of attrition scenario.