Second-Price Sealed-Bid Auction Consider 2 individuals who are interested in one indivisible object. Each player $i$ has a valuation $v_i &gt; 0$ 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

$$u_1(b_1, b_2) =  \left\{
  \begin{array}{l l}
    v_1 - b_2 &amp; \quad \text{if $b_1 \geq b_2$}\\
    0 &amp; \quad \text{if $b_1 &lt; b_2$}
  \end{array} \right. $$ and 
$$u_2(b_1, b_2) =  \left\{
  \begin{array}{l l}
    v_2 - b_1 &amp; \quad \text{if $b_2 &gt; b_1$}\\
    0 &amp; \quad \text{if $b_2 \leq b_1$}
  \end{array} \right. $$

Am I correct if I say that in this case, the best reply functions are given by:

$$\beta_1(b_2) =  \left\{
  \begin{array}{l l}
    \{b_1 | b_1 \geq b_2\} &amp; \quad \text{if $b_2 &lt; v_1$}\\
   \{b_1 | b_1 &lt; b_2\}  &amp; \quad \text{if $b_2 &gt; v_1$} \\
[0, \infty]  &amp; \quad \text{if $b_2 = v_1$}
  \end{array} \right. $$

and

$$\beta_2(b_1) =  \left\{
  \begin{array}{l l}
    \{b_2 | b_2 &gt; b_1\} &amp; \quad \text{if $b_1 &lt; v_2$}\\
   \{b_2 | b_2 \leq b_1\}  &amp; \quad \text{if $b_1 &gt; v_2$} \\
[0, \infty]  &amp; \quad \text{if $b_1 = v_2$}
  \end{array} \right. $$
These look right to me.  Why did you doubt them?
The second price auction has a dominant strategy equilibrium in which each player bids their true valuation. Given the other player bids $b_{-i}$, consider the 3 possible cases for player $i$:


$v_i &gt; b_{-i}$. Player $i$ should bid any $b_i \in (b_{-i}, \infty)$ and get payoff $v_i - b_{-i}$. Otherwise he gets $0$.
$v_i = b_{-i}$. Player $i$ is indifferent between any bids.
$v_i &lt; b_{-i}$. Player $i$ should bid any $b_i \in (-\infty, b_{-i})$ and get payoff $0$. Otherwise he gets negative payoff.