References for this game - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:05:01Z http://mathoverflow.net/feeds/question/92673 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92673/references-for-this-game References for this game Matteo Mio 2012-03-30T11:12:55Z 2012-04-01T18:14:48Z <p>Hello everybody, </p> <p>I would like to know how the following game is known in the literature and, possibly, to have references for related papers.</p> <p><strong>Description of the game:</strong> Fix a space $X$ and two Borel probability measures $\mu$ and $\nu$ over $X$. There are two players, $A$ and $B$. They both know $\mu$ and $\nu$. Player $A$ chooses between $\mu$ and $\nu$. Player $B$ can not observe Player $A$'s choice. Say $\mu$ is chosen by $A$. Then an element $x\in X$ is randomly chosen in accordance with $\mu$. Now player $B$, looking at $x$, must guess the choice of $A$, i.e., Player $B$ must say "you piked $\mu$", or "you picked $\nu$". Player $B$ wins if their guess is correct. Player $A$ wins otherwise.</p> <p><strong>How to formalize the game:</strong></p> <ol> <li><p>The strategies for Player $A$ can be formalized as (randomized) choices over the two element set, i.e., as elements in $[0,1]$.</p></li> <li><p>A strategy for Player $B$ can be formalized as a map $\sigma: X\rightarrow [0,1]$: if $x$ is the outcome, then guess $\mu$ with probability $\sigma(x)$ and $\nu$ with probability $1-\sigma(x)$. </p></li> <li><p>Since Player $B$ can not observe Player $A$'s choice, the game can be consider as played concurrently.</p></li> </ol> <p>I believe the game has an optimal equilibrium and its value is a function of $\displaystyle \bigsqcup_{B\ Borel} | \mu(B) - \nu(B) |$, i.e., of the total variation distance between $\mu$ and $\nu$.</p> <p>Thank you in advance for any information.</p> <p>Matteo</p>