Consider the following 'war of attrition'. Two animals are in a stand off for a prey....

 

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Consider the following 'war of attrition'. Two animals are in a stand off for a prey. They independently decide when to give up. Waiting is costly, but the animal giving up last wins the prey (they each get nothing if they walk away at the exact same time). Getting the prey gives a benefit of 80 while waiting costs 2 per unit of time. Formally payoffs are given as follows: u₁(t₁, t₂) = u₂(t₁, t₂) = S-2t₁ 802t2 80-2t₁ -2t₂ if t₁ < t₂ if t₁>t₂ if t₁ < t₂ if t₁ ≥ t₂, where t, is the amount of time animal i decided to wait. Assuming that animals aim to maximize payoffs (consciouly or not), figure out the Nash equilibria of this game by answering the following questions (similar to how we proceeded to solve the Bertrand game). (a) Show that there is no Nash equilibrium where both animals wait a strictly positive amount of time. For this, consider two subcases: (i) both wait the same amount of time, or (ii) one gives in earlier than the other. (b) Assume now that one animal, say the first one, gives up right away (t₁ = 0) while the other picks t2 20. For which values of t2 do we have a Nash equilibrium? (c) Are the Nash equilibria identified in (b) Pareto efficent? Consider the following 'war of attrition'. Two animals are in a stand off for a prey. They independently decide when to give up. Waiting is costly, but the animal giving up last wins the prey (they each get nothing if they walk away at the exact same time). Getting the prey gives a benefit of 80 while waiting costs 2 per unit of time. Formally payoffs are given as follows: u₁(t₁, t₂) = u₂(t₁, t₂) = S-2t₁ 802t2 80-2t₁ -2t₂ if t₁ < t₂ if t₁>t₂ if t₁ < t₂ if t₁ ≥ t₂, where t, is the amount of time animal i decided to wait. Assuming that animals aim to maximize payoffs (consciouly or not), figure out the Nash equilibria of this game by answering the following questions (similar to how we proceeded to solve the Bertrand game). (a) Show that there is no Nash equilibrium where both animals wait a strictly positive amount of time. For this, consider two subcases: (i) both wait the same amount of time, or (ii) one gives in earlier than the other. (b) Assume now that one animal, say the first one, gives up right away (t₁ = 0) while the other picks t2 20. For which values of t2 do we have a Nash equilibrium? (c) Are the Nash equilibria identified in (b) Pareto efficent?

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a Lets consider the two subcases i Both wait the same amount of time If both animals wait the same a... View the full answer