Solving 2Xn or mX2 Games by Graphical Method

GAME THEORY : SOLVING 2Xn OR mX2 GAMES

When is this used •

•

•

•

When either player A or player B has only 2 strategies and the other player has more than 2 strategies No saddle point is found 2 X n game means: player A has only 2 strategies and Player B has n strategies m X 2 game means: player A has m strategies and Player B has only 2 strategies

Algorithm for 2 X n game •

•

•

•

•

Step 1 : Reduce the payoff matrix by applying dominance property, if it exists Step 2: Let, x be the probability for Player A taking alternative 1, then (1x) becomes the probability for alternative 2. Derive the expected gain function of player A Step 3 :For each gain function find values when x = 0 & x=1 Step 4 : plot gain. Keep Keep x value on X axis and gain on Y axis Step 5: Since player A is a maximin player, find the highest intersection point in the lower boundary of the

(contd..)

•

•

Step 6 : If the no. of lines passing through the maximum point is only two, from a 2 x 2 payoff matrix from the original problem by retaining only the columns corresponding to those two lines and go to step 8, otherwise go to step 7 Step 7: If more than two lines are passing through the maximum point, identify any two lines with opposite slopes passing through that point.  Then form 2x2 payoff matrix from the original problem by retaining only the columns corresponding to those two lines which are having opposite slopes Step 8 :Solve the 2x2 game with

Solve the below 2 x 5 game graphically Playe Player r B 1 1 Player Player A 2

5

3 -1

2

0 5

3

6 -2

-1 2

4

7 1

Solving 1 1 Play Player er A 2 Column maximum 2

3 -1

0 5 3 7

Player Player B 2 4

6 -2

-1 2

3 5

Row minimum minimum -1 (Maxim aximi in) -2

7 1

5

(Minimax) Maximin = Minimax. No No saddle saddle point point

6

Checking for dominancy in column (as only 2 rows are present) Player Player B 1

2

3

4

3 -1

1 Play Player er A

0 5

5

6 -2

-1 2

7 1

2 Column Column dominanc dominancy y: Colu Column mn 4 cell cell valu values es are are < colu column mn 2 valu values es. So colu column mn 4 is domi domina nant nt. Dele Delete te colu column mn 2 Column Column dominanc dominancy y: Colu Column mn 43 cell cell valu values es are are < colu column mn 5 valu values es. So colu column mn 3 is domi domina nant nt. Delete Delete colum column n 5 After de deleting co column 2 & 5 Playe Player r B 1 4

1 Playe Player r A 2

3 -1

3

6 -2

-1 2

Now derive the gain function Playe Player r B

1

3

4 1 Player Player A 2

3 -1

B’s alternative 1 3 4

6 -2

-1 2

A’s expected payoff function 3x + (-1 * (1-x) ) = 3x-1+x = 4x-1 6x + (-2 * (1-x) ) = 6x -2 +2x = 8x-2 (-1 * x ) + (2* (1-x) ) = -1x +2 – 2x = -3x +2

Now compute gain for x = 0 & x=1 B’s A’s alternativ expected e  payoff  function 1 4x-1 3 8x-2 4 -3x +2

A’s expected gain X=0

X=1

-1 -2 2

3 6 -1

Now plot the graph

Play Player er A maxi maximi min n play player er, shad shade e the the lowe lowest st boun bounda dary ry of the the grap graph h. Inter tersect secti ion po point ints are are a ,b,c & d . High ighest est poi point is c . Opti ptimal mal solut lution ion exis xists at c. Che Check the no . of line lines s pass passin ing g thru thru c . 2 line ines . So cons consid ide er the the corr corre espon spond ding ing col columns mns (B1 & B4 ) to ma make 2X2 2X2 matr matrix ix

Now we have got 2X2 matrix and can solve through oddments Playe Player r B 1 4 1 Play Player er A 2

3 -1

Oddments

-1 2

Oddments 3 4

3 4

Calculating Calculating probability probability P1= 3/7, P2 = 4/7, q1 = 3/7 Calculating value V /7 = 5/7

& q2 = 4/7

= (3*3) + (-1 *4) / (3 +4 ) = (9 -4 )