Solution to Card Game with Bluffing

There are two players: $A$ and $B$.

First it is $A$'s move.

$A$ receives one of two possible cards: either the high card or the low card.

If he receives the high card, he must bid $2$ credits.

If he receives the low card, he has two options, either:

$\map A 1: \quad$ He may pay $1$ credit, and the play is complete.

or

$\map A 2: \quad$ He may bid $2$ credits.

If $A$ has bid $2$ credits, it is $B$'s move.

$B$ has two options, either:

$\map B 1: \quad$ He may pay $1$ credit

or

$\map B 2: \quad$ He may challenge $A$'s bid.

If $A$ had the high card, $B$ must pay $2$ credits to $A$.

If $A$ had the low card, $A$ must pay $2$ credits to $B$.

The play is complete.

Proof

From the payoff table:

		$\text B$
$\text A$		$\begin{array} {r {{\|}} c {{\|}} c {{\|}} } & B_1 & B_2 \\ \hline A_1 & 1/2 & 0 \\ \hline A_2 & 0 & -1 \\ \hline \end{array}$

The solution is:

$A$ takes strategy $A_1$ for $2/3$ of the time, and $A_2$ for $1/3$ of the time.

$B$ takes strategy $B_1$ for $2/3$ of the time, and $B_2$ for $1/3$ of the time.

This article contains statements that are justified by handwavery.
In particular: It is not made clear in the source work why this is. You just get: "This is indeed a solution, because no player can do better if the other sticks to his strategy, but could gain or lose more than $1/3$ (which is what $A$ gains now) if he departed from the optimal solution and his opponent took advantage of it." This is handwaving. A source work with a higher level of precision is needed here.
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Sources

1956: Steven Vajda: The Theory of Games and Linear Programming ... (previous) ... (next): Chapter $\text{I}$: An Outline of the Theory of Games: $3$

Solution to Card Game with Bluffing