Star (game theory)

<h2 id="why---0">Why ∗ ≠ 0</h2>
<p>A <a href="/facts/Combinatorial_game/fF4l0om2">combinatorial game</a> has a positive and negative player; which player moves first is left ambiguous. The combinatorial game <a href="/facts/Zero_(game)/5Gvqy7pR">0</a>, or { | }, leaves no options and is a second-player win. Likewise, a combinatorial game is won (assuming optimal play) by the second player <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> its value is 0. Therefore, a game of value ∗, which is a first-player win, is neither positive nor negative. However, ∗ is not the only possible value for a first-player win game (see <a href="/facts/Nimber/jjW5SVmm">nimbers</a>).
</p><p>Star does have the property that the <a href="/facts/Sum_of_combinatorial_games/qS2BARP9">sum</a> ∗ + ∗, has value 0, because the first-player's only move is to the game ∗, which the second-player will win.
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<h2 id="example-of-a-value--game">Example of a value-∗ game</h2>
<p><a href="/facts/Nim/JfDdzxyc">Nim</a>, with one pile and one piece, has value ∗. The first player will remove the piece, and the second player will lose. A single-pile Nim game with one pile of <i>n</i> pieces (also a first-player win) is defined to have value ∗<i>n</i>. The numbers ∗<i>z</i> for <a href="/facts/Integer/PUwwrawx">integers</a> <i>z</i> form an infinite <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> 2, when addition is defined in the context of combinatorial games and multiplication is given a more complex definition.
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Nimber/jjW5SVmm">Nimbers</a></li>
<li><a href="/facts/Surreal_number/VUHPpTKr">Surreal numbers</a></li></ul>

<ul><li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, J. H.</a>, <i><a href="/facts/On_Numbers_and_Games/1HRjcfA1">On Numbers and Games</a>,</i> <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a> Inc. (London) Ltd., 1976</li></ul>

Star (game theory) open-in-new

Star (game theory)