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Consider a two-player first-price auction with complete information, in which players' values $V_2\geq V_1$ for the prize are commonly known. If the players tie, the winner is determined randomly (each player with probability 0.5). Suppose players bid sequentially and bids are observable.

Why are there no subgame perfect equilibria whoever bids first? The solution explains it like this:

In the subgame where the first bidder bids $b<V_1$, the second bidder's best response is empty. So in this subgame there is no Nash equilibrium. By definition, a SPE has to be NE in every subgame, so there is no SPE in this game.

I'm having trouble understanding this.

Also, if players bid simultaneously, what is a mixed-strategy Nash equilibrium?

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    $\begingroup$ The idea is that there is no such thing as a smallest bid that is strictly larger than $b$. $\endgroup$ Apr 4 at 22:03
  • $\begingroup$ @MichaelGreinecker except that in real life monetary bids are discrete, thanks to cents or pennies or auctioneer rules or whatever. $\endgroup$
    – Henry
    Apr 4 at 22:10
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    $\begingroup$ @Henry And you are telling me that why? $\endgroup$ Apr 4 at 23:24
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    $\begingroup$ @Henry You seem to be acting under the misguided assumption that this proof/exercise has something to do with real life? $\endgroup$
    – Giskard
    Apr 5 at 3:36
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    $\begingroup$ "Also, if players bid simultaneously, what is a mixed-strategy Nash equilibrium?" is a very different question, should be posted seperately or even better, you can just edit it out and find an older question that already asked it. $\endgroup$
    – Giskard
    Apr 5 at 5:50

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