# Finding the Bayes-Nash Equilibrium for First-Price Auction with 2 bidders

Let $$\sigma_i$$ be the strategy profile for bidder $$i$$ that indicates how they should bid based on their value. As we know, if there are 2 bidders both with their values $$v_1,v_2$$ on $$U[0,1]$$ in a First-Price Auction, they will bid $$\frac{v_1}{2}$$ and $$\frac{v_2}{2}$$ respectively, so $$\sigma_1(v_1)=\frac{v_1}{2},\sigma_2(v_2)=\frac{v_2}{2}$$.

However, if bidder 1's value is still $$U[0,1]$$ but bidder 2's value is $$U[0,2]$$, there does not exist a Bayes-Nash Equilibrium. Why is this? Specifically, why is the following not a BNE?

$$\begin{equation*} \begin{split} \sigma_1(v_1) &= \frac{v_1}{2} \\ \sigma_2(v_2)&= \begin{cases} \frac{v_2}{2} & \mbox{ if }v_2\in[0,1] \\ \frac{1}{2} & \mbox{ if }v_2>1 \end{cases} \end{split} \end{equation*}$$

Equilibrium says for all bidders $$i$$ and values $$v_i$$, $$\sigma_i(v_i)$$ is the optimal bid. Is there a bidder and value for which this is not the case?

It is easy to see why the strategy profile does not form a BNE. Since bidder 2 never bids above $$\frac12$$, for a sufficiently high $$v_1$$, an alternative strategy $$\sigma_1'(v_1)=\frac12+\epsilon$$ is a better response to $$\sigma_2$$ than the proposed $$\sigma_1$$.
For example, suppose bidder 1 observes $$v_1=1$$. Following $$\sigma_1$$, his probability of winning is $$\Pr(\sigma_2\le \frac12)=\frac12$$ and so his expected payoff is $$\frac12(1-\frac12)=\frac14$$. But if he bids slightly above $$\frac12$$, his probability of winning jumps to $$1$$, and his expected payoff is just slightly below $$\frac12$$, which is greater than $$\frac14$$.
In fact, there does exist a BNE in the asymmetric two-bidder first price auction where values are uniformly distributed on $$[0,\omega_i]$$, with $$\omega_1\ne \omega_2$$. The equilibrium bidding strategy is given by $$$$\sigma_i(v_i)=\frac{1}{k_iv_i}\left(1-\sqrt{1-k_iv_i^2}\right),\qquad\text{where }k_i=\frac1{\omega_i^2}-\frac1{\omega_j^2}.$$$$ For the details of derivation, I'd refer you to Section 4.3 of Krishna (2010).