Question

Please show the answer to this problem in Microsoft Excel using linear programming and solver with all work and screenshots. Please provide the relevant game theory table as well.

John and Smith are two colleagues that play the following game in the lunch room of their job. John picks an integer between 1 and 4, and Smith picks an integer between 3 and 6. If the sum of the two numbers equals 7, Smith wins $8 from John. If the sum of the two numbers equals 6, John wins $6 from Smith. If the sum of the two numbers equals 9, John wins $9 from Smith. If the sum of the numbers is anything other than 6, 7, or 9, the game is a tie and neither player wins or loses. Determine the strategies of each player and the expected value of the game. If the game is played for 100 rounds, what is the expected outcome?

Answer 1

The question can be formulated as shown below:

Smith 4 Max(Min) Probability Strate 4 P1 P2 P3 P4 4 John 4 Min(Max) 6 10

From above it is clear that, the overall value of the game lies between -8 & 6.

Now,

Objective function of the LP with respect to Smith will be : Minimize V

Now, formulate the LP model as per the strategies being chosen as shown below:

Now, if John select the strategy 1 i.e. if John chosen 1 then the value of the Smith will be

(q1X0 + q2X0 + q3X5 + q4X-8) <= V

Similarly if John selects strategy 2, 3 & 4 i.e. if he selects 2, 3 &4 respectively then value of Smith will be:

(q1X0 + q2X6 + q3X-8 + q4X0) <= V

(q1X6 + q2X-8 + q3X0 + q4X9) <= V

(q1X-8 + q2X0 + q3X9 + q4X0) <= V

Also, q1+q2+q3+q4 = 1 ...........(i)

Now, dividing all the above 4 equations by V to both side and considering qi/V = yi , where i = 1,2,3,4 then the equations will be:

$y_{1}\times 0+y_{2}\times 0+y_{3}\times 6+y_{4}\times -8 \leq 1$

$y_{1}\times 0+y_{2}\times 6+y_{3}\times -8+y_{4}\times 0 \leq 1$

$y_{1}\times 6+y_{2}\times -8+y_{3}\times 0+y_{4}\times 9 \leq 1$

$y_{1}\times -8+y_{2}\times 0+y_{3}\times 9+y_{4}\times 0 \leq 1$

And from Eq ..(i),

Now, Minimizing V means, Maximizing Means Maximizing $y_{1}+y_{2}+y_{3}+y_{4}$

subject to Constraints

$y_{1}\times 0+y_{2}\times 0+y_{3}\times 6+y_{4}\times -8 \leq 1$

$y_{1}\times 0+y_{2}\times 6+y_{3}\times -8+y_{4}\times 0 \leq 1$

$y_{1}\times 6+y_{2}\times -8+y_{3}\times 0+y_{4}\times 9 \leq 1$ and

$y_{1}\times -8+y_{2}\times 0+y_{3}\times 9+y_{4}\times 0 \leq 1$

Now, solving this LP model as shown below using Excel:

1.54595588235294 2.14705882352941 1.48529411764706 0.988970588235294 Variables Objective Function -SUM(C2:F2) 10 12 13 14 15 16 17 Constraints SUMPRODUCT(C2:F2,C8:F8)< =SUMPRODUCT(C2:F2,C9:F9) <= =SUMPRODUCT(C2:F2,C10:F10) < =SUMPRODUCT(C2:F2,C11:F11) <= -D2 19 -F2 21

The solution of this LP model will be as shown below:

Now, from the Eq ....(i),

This means, 1.55 +2.15 + 1.49 + 0.99 = 1/V

=> V = 1/6.1672 = 0.162

Hence the Value of the Game is 0.162