Information set (game theory)

<h2 id="example">Example</h2>

<p>At the right are two versions of the <a href="/facts/Battle_of_the_sexes_(game_theory)/Y1pRYk21">battle of the sexes</a> game, shown in <a href="/facts/Extensive_form_game/Yw9gQ7um">extensive form</a>. Below, the <a href="/facts/Normal-form_game/1zxxyHC0">normal form</a> for both of these games is shown as well.
</p><p>The first game is simply sequential―when player 2 makes a choice, both parties are already aware of whether player 1 has chosen O(pera) or F(ootball).
</p><p>The second game is also sequential, but the dotted line shows player 2's information set. This is the common way to show that when player 2 moves, he or she is not aware of what player 1 did.
</p><p>This difference also leads to different predictions for the two games. In the first game, player 1 has the upper hand. They know that they can choose O(pera) safely because <i>once player 2 knows</i> that player 1 has chosen opera, player 2 would rather go along for o(pera) and get 2 than choose f(ootball) and get 0. Formally, that's applying <a href="/facts/Subgame_perfect_equilibrium/cyyhua3F">subgame perfection</a> to solve the game.
</p><p>In the second game, player 2 can't observe what player 1 did, so it might as well be a <a href="/facts/Simultaneous_game/TkHcpJUo">simultaneous game</a>. So subgame perfection doesn't get us anything that <a href="/facts/Nash_equilibrium/l74xzIHd">Nash equilibrium</a> can't get us, and we have the standard 3 possible equilibria:
</p>
<ol><li>Both choose opera</li>
<li>both choose football</li>
<li>or both use a <a href="/facts/Mixed_strategy/VEXD1jyj">mixed strategy</a>, with player 1 choosing O(pera) 3/5 of the time and choosing football 2/5 of the time, and player 2 choosing f(ootball) 3/5 of the time and opera 2/5 of the time</li></ol>

<table><tbody><tr><td><i>Normal form with player 2 aware of player 1's move</i><table><tbody><tr><th>Player 2Player 1</th><th>Oo, Fo</th><th>Oo, Ff</th><th>Of, Fo</th><th>Of, Ff</th></tr><tr><th>O</th><td> 2 3 </td><td> 2 3</td><td> 0 0</td><td> 00 </td></tr><tr><th>F</th><td> 0 0</td><td> 32 </td><td> 00 </td><td> 3 2 </td></tr></tbody></table></td><td><i>Normal form with player 2 unaware of player 1's move</i><table><tbody><tr><th>Player 2Player 1</th><th>o</th><th>f</th></tr><tr><th>O</th><td> 2 3 </td><td> 00 </td></tr><tr><th>F</th><td> 0 0</td><td> 3 2 </td></tr></tbody></table></td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Self-confirming_equilibrium/m6I0S85D">Self-confirming equilibrium</a></li></ul>

<ul><li><a href="/facts/Ken_Binmore/0RJX9nw5">Binmore, Ken</a> (2007). <a href="https://books.google.com/books?id=hRB34xDNc2AC&pg=88"><i>Game Theory: A very short introduction</i></a>. Oxford University Press. pp. 88–89. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-921846-2.</li></ul>

Information set (game theory) open-in-new

Information set (game theory)