According to the answer key for a problem set, there is no pure strategy Nash equilibrium in the following problem. Yet I can't see why not. Could it be an error in the answer key? Here's the problem:

"Consider two risk-neutral buyers that submit bids to buy an object whose value v is either 0 or 1.

  • The informed buyer $I$ knows whether v = 1 or v = 0.
  • The uninformed buyer $U$ assigns $Pr(v = 1)=Pr(v=0)=1/2$

The high price buyer wins the object at a price equal to the high price."

I don't understand why the following strategy profile $\{P_I^*, P_U^*\}$ is not a pure Nash equilibrium (where $P_i^*$ is the bid submitted by $i\in \{I, U\}$):

\begin{equation*} P_I^* = \begin{cases} 0 \quad \text{if}\quad v=0 \\ 1 \quad \text{if}\quad v=1 \end{cases} \end{equation*} \begin{equation*} P_U^* = \begin{cases} 0 \quad \text{if}\quad P_I=0 \\ 1 \quad \text{if}\quad P_I\neq 0 \end{cases} \end{equation*}

Am I right that this is a pure NE? If not, can someone help me see why it is not? Thanks in advance!

[Here is why I think it is a pure NE:

  • If $v=0$, then Buyer $I$ cannot gain by deviating from $P_I^*$ (because that would mean losing the bid given $U$'s strategy unless both bids are 1 in which case there is no profit anyway)
  • If $v=1$, then Buyer $I$ cannot gain by deviating because that would mean losing the bid given $U$'s strategy.
  • If Buyer $U$ submits a price higher than $P_I$ when $P_I=0$, then $U$ will win the bid and lose money (because in that case $v=0$).
  • If Buyer $U$ submits a price lower than $P_I$ when $P_I=1$, then $U$ loses the bid and gets nothing.]
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    $\begingroup$ The uninformed buyer cannot condition her strategy on the outcome of the informed player, as they move simultaneously (i.e. she does not know what was played by the informed party before she makes her bid). $\endgroup$ – Walrasian Auctioneer May 16 at 1:32
  • $\begingroup$ Thanks for the reply! In a NE, the player does what is best given the other players strategy. Shouldn’t that imply that both players in a sense ”observe” each others bids and adjust their bids accordingly? $\endgroup$ – Nomista May 16 at 1:43
  • $\begingroup$ E.g. that seems to be the reasoning in the answer key: ”Suppose to the contrary and let b denote the bid of "U" in the pure NE. Now, it is optimal that an "I" submits a bid slightly above b when the value is high (and to submit a bid equal to zero otherwise). Thus, "U" makes a loss with the pure strategy b. [A similar argument can be given for I.]” $\endgroup$ – Nomista May 16 at 1:45
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    $\begingroup$ Yes, U knows the strategy I uses, but a pure strategy for U is just a number $b$. Contrast this with I, who chooses a bid depending on $v$. $\endgroup$ – Walrasian Auctioneer May 16 at 5:00

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