This is an all-pay auction (Highest bidder wins the object, all players pay what they bid, player 1 wins all ties): Player 1 has $300$ dollars, Player 2 has $500$ dollars, the object being auctioned is $100$ dollar bill. The two players only care about the amount of cash they carry at the time the auction ends.
Player 1 has utility $u_1 (x) = x^{1/2}$, Player 2 has utility $u_2 (x) = ln (x)$ for $x$ dollars. Player 1 bids between $[0,300]$, player 2 bids between $[0,500]$.
Find a mixed-strategy Nash equilibrium.
I know that I'm supposed to eliminate all strictly dominated strategies first, and I think Player 2 has $(300,500]$ as strictly dominated strategy. But I'm stuck to find the mixed strategy Nash equilibrium. Could someone help me with this please? Thanks.