A Cooperative Dual to the Nash Equilibrium  for Two-Person Prescriptive Games

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr class="thead"><td align="left"> </td><td align="center"> </td><td align="center" colspan="3">Player II</td><td align="center"> </td></tr><tr class="thead"><td align="left"> </td><td align="center"> </td><td align="center"><span style="width: 12.25px;"><svg height="14.2625" id="M140" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 12.25 14.2625" width="12.25" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,5.862,14.2)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg></span></td><td align="center"><span style="width: 12.25px;"><svg height="14.2625" id="M141" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 12.25 14.2625" width="12.25" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.125)"><path d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" id="x1D461"></path></g>
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</svg></span></td><td align="center"><span style="width: 12.25px;"><svg height="14.4125" id="M142" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 12.25 14.4125" width="12.25" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,5.862,14.2)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
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</svg></span></td><td align="center">max</td></tr><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr><td align="left" rowspan="2">Player I</td><td align="center"><span style="width: 12.8375px;"><svg height="11.825" id="M143" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 12.8375 11.825" width="12.8375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81
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<g transform="matrix(.012,-0,0,-.012,6.438,11.762)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg></span></td><td align="center">5</td><td align="center">2</td><td align="center"><b>6</b></td><td align="center">6</td></tr><tr><td align="center"><span style="width: 12.8375px;"><svg height="11.825" id="M144" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 12.8375 11.825" width="12.8375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg></span></td><td align="center">3</td><td align="center">6</td><td align="center">7</td><td align="center">7</td></tr><tr><td align="left"> </td><td align="center">min</td><td align="center">3</td><td align="center">2</td><td align="center">6</td><td align="center"> </td></tr><tr class="table-tr"><td colspan="6"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Journal of Applied Mathematics

tab4

Table 4

Table 4: A Cooperative Dual to the Nash Equilibrium  for Two-Person Prescriptive Games 