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Let we think about $textbfFirst Price Sealed Bid Auction$ in which bidder who offers highest bid wins the object. There are only two bidders in this auction for the simplicity, $I_1$ and $I_2$.

Bids are strategies for the two bidders and bid strategies for two bidders are continious not discrete. Bids are monetary/dollar.

Each bidders valuation for this object is $v_i$ where $i=1,2$. For the simplicity, $v_1>v_2$.

Payoff functions for each bidder is $u_i=(v_i-b_i)$.

Tie breaking rule for this auction is: When two bidders offer same bid $b_1=b_2=b$, then their payoffs are $u_i=dfrac(v_i-b)2$ for $i=1,2$.

The information structure is complete. Namely, valuations and bids are common knowledge for two bidders.

Now, it can be easily seen that there is no pure Nash Equilibrium under these game structure with continious bidding since, in any case, there is a profitable deviation incentive at least for one bidder if the bids are between $v_2leq b_2leq b_1 leq v_1$ or $v_2leq b_1leq b_2 leq v_1$. I do not mention the case in which two bids are above $v_1$ since this is trivial that there is no any type of Nash Equilibrium above this valuation.

My question about the analysis for the mixed stratgegy Nash equilibria. The important point is that since the information structure is complete, there is no type on bidders which any bidder should appoint a belief to the other bidder. For that reason, we cannot find Bayes-Nash Equilibrium. In this regard, I need a help for $textMixed Strategy Nash Equilibria$ analysis in First Price Sealed Bid Auction with Complete Information under these game rules.