1. Tools for Decision Analysis: Analysis of Risky Decisions
If you will begin with certainties, you shall end in doubts, but if you will content to begin with doubts, you shall end in almost certainties.

2. ______________________________________
Introduction & Summary
Game theory describes the situations involving conflict in which the payoff is affected by the actions and counter-actions of intelligent opponents. Two-person zero-sum games play a central role in the development of the theory of games.
Game theory is indeed about modeling for winning business in a competitive environment: For example, in winning a large bid, there are factors that are important. These factors include: Establishing and maintaining a preferred supplier position, developing a relationship of trust with the customer, the offering itself, and the price.
In order to develop the game theory concepts, consider the following game in which player I has two choices from which to select, and player II has three alternatives for each choice of player I. The payoff matrix T is given below:
player II
j=1
j=2
j=3
player I
i=1 4 1 3
i=2 2
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The Payoff Matrix

3. In the payoff matrix, the two rows (i = 1, 2) represent player I strategies , and the three columns (j = 1, 2, 3) represent the player II strategies . The payoff matrix is oriented to player I, meaning that a positive tij is a gain for player I and a loss for player II, and a negative tij is a gain for player II and a loss for player I.
For example, if player I uses strategy 2 and player II uses strategy 1, player I receives t21 = 2 units and player II thus loses 2 units. Clearly, in our example player II always loses; however, the objective is to minimize the payoff to player I.
A pure strategy pair (i. j) is in equilibrium if and only if the corresponding element tij is both the largest in its column and the smallest in its row. Such an element is also called a saddle point. An "equilibrium decision point", that is a "saddle point", also known as a "minimax point", represents a decision by two players upon which neither can improve by unilaterally departing from it.
When there is no saddle point, one must choose the strategy randomly. This is the idea behind a mixed strategy
We will use a general method based on a linear programming (LP) formulation. This equivalency of games and LP may be surprising, since a LP problem involves just one decision-maker, but it should be noted that with each LP problem there is an associated problem called the dual LP. The optimal values of the objective functions of the two LPs are equal, corresponding to the value of the game. When solving LP by simplex-type methods, the optimal solution of the dual problem also appears as part of the final tableau

4. LP formulation and the simplex method is the fastest, most practical, and most useful method for solving games with a large matrix T. Suppose that player II is permitted to adopt mixed strategies, but player I is allowed to use only pure strategies. What mixed strategies Y = (y1, y2, y3) should player II adopt to minimize the maximum expected payoff v? A moment's thought shows that player II must solve the following problem:
Min v = y
subject to:
T.Y £ y
Ut.Y = 1
The LP formulation for player II's problem in a game with payoff matrix T given above, is:
Min v
subject to:
4y1 + y2
3y3 v
2y1
3y2
4y3 y1 y3 = 1
yj³0, j = 1, 2, 3, and v is unrestricted

5. The optimal solution for player II is:
y1 = 1/2, y2 = 1/2, y3 = 0.
The shadow prices are the optimal strategies for player I. Therefore, the mixed saddle point is:
x1 = 1/4, x2 = 3/4; y1 = 1/2, y2 = 1/2, y3 = 0,
and the value of the game equals 5/2.
Note that the essential strategies for Player I are i = 1, i = 2;
for Player II they are j = 1, j = 2 while j = 3 is non-essential.

6. States of Nature
Growth
Medium G
No Change
Low G MG N L
Actions
Bonds
12% 8 7 3
Stocks 15 9 5 -2
Deposit
In decision analysis, the decision-maker has to select at least and at most one option from all possible options. This certainly limits its scope and its applications. You have already learned both decision analysis and linear programming. Now is the time to use the game theory concepts to link together these two seemingly different types of models to widen their scopes in solving more realistic decision-making problems. The investment problem can be formulated as if the investor is playing a game against nature.

7. Suppose our investor has $100,000 to allocate among the three possible investments with the unknown amounts Y1, Y2, Y3, respectively. That is,
Y1 + Y2 + Y3 = 100,000
Notice that this condition is equivalent to the total probability condition for player I in the Game Theory.
Under these conditions, the returns are:
0.12Y1 +
0.15Y2
0.07Y3
{if Growth (G)}
0.08Y1
0.09Y2
{if Medium G}
0.07Y1
0.05Y2
{if No Change}
0.03Y1 -
0.02Y2
{if Low}

8. The objective is that the smallest return (let us denote it by v value) be as large as possible.
Formulating this Decision Analysis problem as a Linear Programming problem, we have:
Max v
Subject to: Y1 + Y2 Y3 =
100,000
0.12Y1
0.15Y2
0.07 Y3 v
0.08Y1
0.09Y2
0.07Y3
0.07Y1
0.05Y2
0.03Y1 -
0.02Y2
and Y1, Y2, Y3 ³ 0, while v is unrestricted in sign (could have negative return).

9. This LP formulation is similar to the problem discussed in the Game Theory section. In fact, the interpretation of this problem is that, in this situation, the investor is playing against nature (the states of economy).
Solving this problem by any LP solution algorithm, the optimal solution is
Y1 = 0, Y2 = 0, Y3 = 100,000, and v = $7000.
That is, the investor must put all the money in the money market account with the accumulated return of
100,000´1.07 = $10,7000.
Note that the pay-off matrix for this problem has a saddle-point; therefore, as expected, the optimal strategy is a pure strategy. In other words, we have to invest all our money into one portfolio only.

10. States of Nature (Economy)
Buying Gold or Foreign Currencies Investment Decision: As another numerical example,
consider the following two investments with the given rate of returns. Given you wish to invest $12,000 over a period of one year, how do you invest for the optimal strategy?
States of Nature (Economy)
Growth
Medium G
No Change
Low G MG N L
Actions
Buy Currencies (C) 5 4 3 -1
Buy Gold (G) 2

11. The objective is that the smallest return (let us denote it by Z value) be as large as possible.
Similar to previous example, formulating this Decision Analysis problem as a Linear Programming problem, we have:
Maximize Z
Subject to: X1 + X2 =
12,000
0.05X1
0.02X2 Z
0.04X1
0.03X2
0.03X1
0.04X2
-0.01X1
0.05X2
and X1, X2 ³ 0, while Z is unrestricted in sign (i.e., could have negative return).

12. Again, this LP formulation is similar to the problem discussed in the Game Theory section.
In fact, the interpretation of this problem is that, in this situation, the investor is playing against nature
(i.e., the states of economy).
Solving this problem by any LP solution algorithm, the optimal solution is a mixed strategy:
Buy X1 = $4000 Foreign Currencies
and X2= $8000 Gold.

13. The Investment Problem Under Risk:
The following table shows the risk measurements computed for the Investment Decision Example:
Risk
Assessment
G(0.4)
MG(0.3)
NC(0.2)
L(0.1)
Exp. Value
St. Dev.
C. V. B 12 8 7 3
8.9
2.9
32% S 15 9 5 -2
9.5*
5.4
57% D 0%
The Risk Assessment columns in the above table indicate that bonds are much less risky than the stocks, while its return is lower. Clearly, deposits are risk free.
Now, an interesting question is: Given all this relevant information, what action do you take? It is all up to you.

14. Max [v Y Y2 -0Y3]
Subject to: Y1 + Y2 Y3 =
100,000
0.089Y1
0.095Y2
0.07Y3 v
and Y1, Y2, Y3 ³ 0, while v is unrestricted in sign (could have negative return).
Solving this Linear Program (LP) model by any computer LP solver, the optimal solution is
Y1 = 0, Y2 = 0, Y3 = 100,000, and v = $7000. That is, the investor must put all the money in the money market account with the accumulated return of 100,000´1.07 = $10,7000.
Notice that, for this particular numerical example, it turns out that the different approaches provide the same optimal decision; however one must be careful not to do any generalization at all.
Note that the above objective function includes the standard deviations to reduce the risk of your decision. However, it is more appropriate to use the covariance matrix instead. Nevertheless the new objective function will have a quadratic form, which can be solved by applying nonlinear optimization algorithms.