<p>Consider a two-player first-price auction with complete information, in which players' values <span class="math-container">$V_2\geq V_1$</span> 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.</p>
<p>Why are there no subgame perfect equilibria whoever bids first? The solution explains it like this:</p>
<blockquote>
<p>In the subgame where the first bidder bids <span class="math-container">$b&lt;V_1$</span>, 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.</p>
</blockquote>
<p>I'm having trouble understanding this.</p>
<p>Also, if players bid simultaneously, what is a mixed-strategy Nash equilibrium?</p>

CC BY-SA 4.0
2024-04-04T22:03:02.803Z
The idea is that there is no such thing as a smallest bid that is strictly larger than $b$.
Michael Greinecker
CC BY-SA 4.0
2024-04-04T22:10:41.807Z
@MichaelGreinecker except that in real life monetary bids are discrete, thanks to cents or pennies or auctioneer rules or whatever.
Henry
CC BY-SA 4.0
2024-04-04T23:24:53.303Z
@Henry And you are telling me that why?
Michael Greinecker
CC BY-SA 4.0
2024-04-05T01:08:02.547Z
Because then there is a smallest bid strictly larger than $b$
Henry
CC BY-SA 4.0
2024-04-05T03:36:44.443Z
@Henry You seem to be acting under the misguided assumption that this proof/exercise has something to do with real life?
Giskard
CC BY-SA 4.0
2024-04-05T05:49:16.600Z
@LudwigGershwin Could you please elaborate on what confuses you in "*the second bidder's best response is empty. So in this subgame there is no Nash equilibrium*"? Do you think the best response exists, or that it is not necessary for an equilibrium to exist, or something else?
Giskard
CC BY-SA 4.0
2024-04-05T05:50:22.860Z
"*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.
Giskard
<game-theory><auctions>
Why is there no subgame perfect equilibrium in a sequential two-player first-price auction?
Ludwig Gershwin
# Why is there no subgame perfect equilibrium in a sequential two-player first-price auction?

(Asked by: username_0 on 2024-04-04)

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, 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?

**Tags:** game-theory, auctions
> **Comments:**
> **username_1**: The idea is that there is no such thing as a smallest bid that is strictly larger than $b$.  
> **username_2**: @MichaelGreinecker except that in real life monetary bids are discrete, thanks to cents or pennies or auctioneer rules or whatever.  
> **username_1**: @Henry And you are telling me that why?  
> **username_2**: Because then there is a smallest bid strictly larger than $b$  
> **username_3**: @Henry You seem to be acting under the misguided assumption that this proof/exercise has something to do with real life?  
> **username_3**: @LudwigGershwin Could you please elaborate on what confuses you in "*the second bidder's best response is empty. So in this subgame there is no Nash equilibrium*"? Do you think the best response exists, or that it is not necessary for an equilibrium to exist, or something else?  
> **username_3**: "*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.  

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## 0 answers

