So I have the following question, dont have much info on class notes and not sure how to tackle it, any suggestions, any help?

Question

A seller has a single item for sale (which she values at zero). There are two potential buyers. The seller decides to use the following auction format to sell the object: each bidder submits a sealed bid; the highest bidder wins in the event of a tie the winner is chosen by a coin toss; the winner pays the average of the two bids. The two bidders are risk-neutral with independent, private values drawn from the Uniform distribution on $[0,1]$. Show that it is a Bayesian Nash equilibrium for each bidder to use the following bidding function:

$$B_i (v_i) =\dfrac{2}{v_i}, \ \ \ \ i = 1, 2$$

Answer 1

If the other guy's valuation is $x$, then he bids $2/x$. We have to check if given the other guys strategy I bid $2/v$ when my value is $v$. Let us define the inicator function $I(b,x)$=[1 if I bid higher, 0 otherwise].

$\\$ Note that this is 1 for $b>(2/x)$ or $x>2/b$.

Expected profit by bidding $b$ is
$ \\= \int_0^1 (v-b/2-1/x)I(b,x)dx \\= \int_0^{\frac2b} (v-b/2-1/x)I(b,x)dx+\int_{\frac2b}^1 (v-b/2-1/x)I(b,x)dx \\=\int_{\frac2b}^1 (v-b/2-1/x)dx \\= (v-b/2)(1-2/b)+log(2/b)$

You can now try differentiating this w.r.t $b$ and then write b as a function of $v$ check if $b^*=2/v$ From what I tried, it was not working out. But, this is how these problems are generally solved.