(Solved): 3. Consider a modified Hawk-Dove/War of Attrition game as follows. Each player chooses a time ti ...

3. Consider a modified Hawk-Dove/War of Attrition game as follows. Each player chooses a time $$t_i$$ to give up waiting. Assume that the value of the prey decreases over time, and its value at time $$t$$ for player $$i$$ is $$v_i?t$$. Upon giving up, the opponent gets the prey and the time at which the winner intended to give up is irrelevant. If they give up at the same time, then the prey is shared (this could also be interpreted as each animal has a $$50\%$$ chance of successfully getting the prey). (a) Write a payoff function $$u_i\left(t_1,t_2\right)$$ for player $$i$$ (separate cases based on the inequality between $$t_i$$ and $$t_j$$ ), sketch a representative graph for the cases $$t_j<v_i,t_j=v_i$$, $$t_j>v_i$$, and box best responses on the graphs. (b) Write the best response $$B_i\left(t_j\right)$$ in piecewise form, graph the best responses in $$\left(t_1,t_2\right)$$ coordinates, and indicate Nash equilibia (graphically is fine). Assume that $$v_1<v_2$$.