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Game Theory

OPERES3 Notes

DefinitionsA game is a generic term, involving conflict situations of particular sort.Game Theory is a set of tools and techniques for decisions under uncertainty involving two or more intelligent opponents in which each opponent aspires to optimize his own decision at the expense of the other opponents. In game theory, an opponent is referred to as player. Each player has a number of choices, finite or infinite, called strategies. The outcomes or payoffs of a game are summarized as functions of the different strategies for each player.

Major Assumptions

1. Players – the number of participants may be two or more. A player can be a single individual or a group with the same objective.

2. Timing – the conflicting parties decide simultaneously.

3. Conflicting Goals – each party is interested in maximizing his or her goal at the expense of the other.

Major Assumptions

4. Repetition – most instances involve repetitive solution.

5. Payoff – the payoffs for each combination of decisions are known by all parties.

6. Information Availability – all parties are aware of all pertinent information. Each player knows all possible courses of action open to the opponent as well as anticipated payoffs.

Classifications of Games

1. Zero-Sum Games – the winner(s) receive(s) the entire amount of the payoff which is contributed by the loser (strictly competitive).

2. Non-Zero Sum Games – the gains of one player differ from the losses of the other. Some other parties in the environment may share in the gain or losses (not strictly competitive).

Two-Person, Zero-Sum Game – Pure Strategy

Characteristics:1.There must be two players, each with a finite set

of strategies.2.Zero-sum implies that the losses of one player is

the exact gain of the other.3.Pure strategy refers to a prescribed solution in

which one alternative is repeatedly recommended to each player.

4.Bargaining is not allowed. There could be no agreement that could be mutually advantageous.


Consider the following game matrix taken from the point of view of player A.

B1 B2 … Bn

A1 v11 v12 … V1n

A2 v21 v22 … v2n...

.

.

.

.

.

.

.

.

.

.

.

.

Am vm1 vm2 … vmn


Example 1:The labor contract between a company and the union will terminate in the near future. A new contract must be negotiated. After a consideration of past experience, the group (Co) agrees that the feasible strategies for the company to follow are:

C1 = all out attack; hard aggressive bargainingC2 = a reasoning, logical approachC3 = a legalistic strategyC4 = an agreeable conciliatory approach

Assume that the union is considering one of the following set of approaches:

U1 = all out attack; hard aggressive bargainingU2 = a reasoning, logical approachU3 = a legalistic strategyU4 = an agreeable conciliatory approach


Example 1 (con’t.)With the aid of an outside mediator, we construct the following game matrix:

Conditional Gains of Union

Company Strategies Union Strategies C1 C2 C3 C4

U1 2.0 1.5 1.2 3.5 U2 2.5 1.4 0.8 1.0 U3 4.0 0.2 1.0 0.5 U4 - 0.5 0.4 1.1 0.0


Example 1 (con’t.)Interpretation of above table or game matrix

If Co. adopts C1 and Union adopts U1, the final contract involves a P2.0 increase in wages (hence, a -P2.0 loss to the company). From the above table, it is clear that if the Company decides to adopt C3, Union will adopt U1. If the Union decides to adopt U3, the company will adopt C2.


Solution Strategy : Minimax – Maximin Approach1. Apply the maximin rule to determine the

optimal strategy for A:

2. Aplly the minimax rule to determine the optimal strategy for B:

[ ]{ }ijijvminmax

[ ]{ }ijijvmaxmin


In the application of the above strategy, the pure strategy problem results in a saddle point, i.e., the payoff corresponding to the maximin rule is identical to the payoff corresponding to the minimax rule.Saddle point corresponds to the minimum in its row and the maximum in its column.


Additional Remarks on Pure Strategy Problems:1.Change in Strategy – Since games are

repetitive, both players may change. But in pure strategy games, there is no incentive to change. Any player deviating from the prescribed strategy will usually find a worsening payoff.

2. Multiple Optimal Solutions – Some games may involve multiple optimal strategies.


Additional Remarks (con’t.)3. Dominance

Row: The dominating row will have entries which are larger than and/or equal to (with at least one entry larger than) to the corresponding entries in the dominated row.Column: The dominating column will have entries smaller than and/or equal to (with at least one entry smaller than) the dominated column

Dominated rows and columns can be deleted from the table.


Example 2: Given game matrix showing the conditional gains of A.

Multiple Pure Strategy Solutions: A2, B2 and A4, B2

B1 B2 B3 minA1 7 -1 2 -1A2 4 4 6 4A3 6 3 0 0A4 7 4 5 4

max 7 4 6


Example 3: Applying Law of Dominance using game matrix of Example 1.

U1 dominates U4. U4 can therefore be removed from the game matrix.After removing U4, we see that C2 dominates C1. C1 can likewise be removed from the table. We are now left with a 3x3 game matrix. This time, we see that U1 dominates both U2 and U3. U2 and U3 can also be removed from the table which leaves us with a 1x3 row vector. Finally, C3 dominates C2 and C4. This leaves us with a single value of 1.2 which corresponds to the value under C3 and U1in the original game matrix. As we already know, C3 and U1 represents the pure strategy solution to this game theory problem.


Example 3: Applying Law of Dominance using game matrix of Example 1.

C1 C2 C3 C4

U1 2.0 1.5 1.2 3.5

U2 2.5 1.4 0.8 1.0

U3 4.0 0.2 1.0 0.5

U4 -0.5 0.4 1.1 0.0

Two-Person, Zero-Sum Game – Mixed Strategy

A mixed strategy problem is one where players change from alternative to alternative when the game is repeated. A mixed strategy problem does not yield a saddle point.


Assumptions in Mixed Strategy Problems:1. The players practice a maximum secrecy with

their plans so that the opponent will not guess their move.

2. The average payoff is determined by the fraction of the time that each of the alternatives is played and there is a certain fraction that is best for each player.

3. The best strategy for a mixed strategy game is a random selection of alternatives which conform in the long run to predetermined proportions.


Example 4: Using Example 1 but replacing (U3,C3) value by 1.9.

C1 C2 C3 C4 Min

U1 2.0 1.5 1.2 3.5 1.2U2 2.4 1.4 0.8 1.0 0.8

U3 4.0 0.2 1.9 0.5 0.2

U4 -0.5 0.4 1.1 0.0 -0.5

max 4.0 1.5 1.9 3.5


Example 4: (con’t.) The intersection of these strategies (U1 and C2) is not an equilibrium or saddle point because 1.5 does not represent both the maximum of its column and the minimum of its row.Interpretation: From the above game matrix, we can see that:

If the Union adopts U1, the Company will adopt C3. If the Company adopts C3, the Union will adopt U3.If the Union adopts U3, the Company will adopt C2.If the Company adopts C2, the Union will adopt U1.The shift from alternative to alternative becomes a cycle when the Union goes back to adopt U1.0


Let xi = proportion of the time that player A plays strategy iyj = proportion of the time that player B plays strategy jPlayer A then selects xi that will yield

=≥ ∑

=

m

iii xx

11,0

∑∑∑===

m

iiin

m

iiii

m

iix

xvxvxvi 11

21

1 ,,,minmax …


Player B then selects yj that will yield:

If xi* and yj

* are the optimal solutions for both players, then the optimal expected value of the game is:

=≥ ∑

=

n

jjj yy

1

1,0

∑∑∑===

n

jjnj

n

jjj

n

jjjy

yvyvyvj 11

21

1 ,,,maxmin …

∑∑= =

=m

i

n

jjiij yxvv

1 1

***


There are several methods for solving this type of game. It is important to first use the principle of dominance to be able to reduce the total number of alternatives. The above non-linear optimization model is convertible to a Linear Programming Model.


Games Reducible to a 2x2 MatrixBy employing the principle of dominance, it may be possible to reduce the size of a game theory problem to a 2x2 matrix.For player A, the optimal strategy involves the simultaneous solution of:

121

222121212111

=++=+

xxvxvxvxvx


Games Reducible to a 2x2 Matrix (con’t)For player B, the optimal strategy involves the simultaneous solution of:

121

222211122111

=++=+

yyvyvyvyvy


Example 5: Using data from Example 4, Reduce the original game matrix using the principle of row and column dominance and determine the mixed strategy solution

C1 C2 C3 C4

U1 2.0 1.5 1.2 3.5

U2 2.4 1.4 0.8 1.0

U3 4.0 0.2 1.9 0.5

U4 -0.5 0.4 1.1 0.0


Solution of (mxn) Games by Linear Programming

As given previously, the following optimization model solves for the optimal strategy of Player A:

∑∑∑===

m

iiin

m

iiii

m

iix

xvxvxvi 11

21

1 ,,,minmax …

ix

xts

i

m

ii

∀≥

=∑=

0

1..1


Solution of (mxn) Games by Linear Programming (con’t.)

This model can be converted to linear programming using the following:

Let

= ∑∑∑

===

m

iiin

m

iii

m

iii xvxvxvv

112

11 ,,,min …



Then, the LP model is given by:

ix

x

jvxvts

vZMax

i

m

ii

m

iiij

∀≥

=

∀≥

=

∑

∑

=

=

0

1

..

1

1



Assuming that v>0, we divide all constraints by v and let Xi=xi/v. Since

, the model for Player A becomes:vv 1minmax ≡

iX

jXvts

Xz

i

m

iiij

m

ii

∀≥

∀≥

=

∑

∑

=

=

0

1..

min

1

1



Using the same principle, player B’s optimization problem is given by:

jY

iYvts

Yw

j

n

jjij

n

jj

∀≥

∀≤

=

∑

∑

=

=

0

1..

max

1

1



Note: In cases where the payoff matrix contains negative payoffs, we scale up all entries by adding a fixed number T which will render all values non-negative. Scaling does not affect the optimal solution except to increase its value by T.


Example 6: Using the game matrix below, find the mixed strategy solution.

y1 y2 y3

x1 6 -4 -14

x2 -9 6 -4

x3 1 -9 1

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