War of Attrition – The Strategy

We have seen the war of attrition in which two players compete for a prize. The price is V, and the cost at each stage is c. In the generic form, let’s define Li as the cumulative cost when i drops out at time, t.

L_i(t) = -c_i – c_id – c_id² … -c_id^t-1
Where d is the discount factor. This factor is required because the game will move into future times. On the other hand, the payoff the player i gets when its opponent drops out at time t is,

H_i(t) = -c_i – c_id – c_id² … -c_id^t-1 + V_i = L_i(t) + V_id^t

There are two pure strategies for this game. They are
1) Player 1 drops out at t = 0, and Player 2 never drops out.
2) Player 2 drops out at t = 0, and Player 1 never drops out.

And the payoffs at the Nash equilibria are: 1) (player 1 = 0 and player 2 = V) and 2) (player 1 = V and player 2 = 0)