Game theory: Consider the first-price sealed-bid auction for the case of uniformly distributed valuations and two bidders under the assumption that the players' strategies are strictly increasing and differentiable. Show that the unique symmetric Bayesian Nash-equilibrium is the linear equilibrium (v₁ /2_, v₂ /2)

Hint: For a given value of v_i, player i's optimal bid solves

max(v_i b_i)Pr[b_i >b(v_j)].

Let b¹(b_j) denote the valuation that bidder j must have in order to bid bj. That is, b¹(b_j) = v_j if b_j = b(v_j). Since v_j is uniformly distributed on [0,1], Pr[b_i > b(v_j)] = Pr[b1(b_i) > v_j] = b¹(b_i). Use the first order condition for player i's optimization problem

to conclude that the unique symmetric Bayesian Nash-equilibrium is (v₁ /2, v₂ /2) .