Consider 2 individuals who are interested in one indivisible object. Each player $i$ has a valuation $v_i > 0$ for the object. Assume $v_1 \geq v_2$. In this scenario, each player submits a bid $b_i \geq 0$. The individual with the highest bid wins the auction and obtains the object at a price equal to the second highest bid, in case of a tie, the object goes to player 1. The payoff functions for each player are given by
$$u_1(b_1, b_2) = \left\{
\begin{array}{l l}
v_1 – b_2 & \quad \text{if $b_1 \geq b_2$}\\
0 & \quad \text{if $b_1 < b_2$}
\end{array} \right. $$ and
$$u_2(b_1, b_2) = \left\{
\begin{array}{l l}
v_2 – b_1 & \quad \text{if $b_2 > b_1$}\\
0 & \quad \text{if $b_2 \leq b_1$}
\end{array} \right. $$
Am I correct if I say that in this case, the best reply functions are given by:
$$\beta_1(b_2) = \left\{
\begin{array}{l l}
\{b_1 | b_1 \geq b_2\} & \quad \text{if $b_2 < v_1$}\\
\{b_1 | b_1 < b_2\} & \quad \text{if $b_2 > v_1$} \\
[0, \infty] & \quad \text{if $b_2 = v_1$}
\end{array} \right. $$
and
$$\beta_2(b_1) = \left\{
\begin{array}{l l}
\{b_2 | b_2 > b_1\} & \quad \text{if $b_1 < v_2$}\\
\{b_2 | b_2 \leq b_1\} & \quad \text{if $b_1 > v_2$} \\
[0, \infty] & \quad \text{if $b_1 = v_2$}
\end{array} \right. $$
Best Answer
These look right to me. Why did you doubt them?