Decision makers are required to make estimations in the decision making process. A method for that is to make an extrapolation of the known circumstances and the past. In order to do this firstly a model is created for data, turned into a hypothesis and then it is assumed that future occurrences will suit this model. The decision maker may realize the actions in a more convenient manner by using the concept of “decision tree” which can produce the best plan by defining the risks and options that may arise in case of risky situations. Decision tree is also called as possibility tree. A decision tree is a schematic representation of any selection problem and shows the possible outcomes of a decision problem and relation of such outcomes with the available actions. Decision trees are assumed to be very popular and useful because of three important reasons. Firstly; this method creates trees which generalize the unobserved examples and define the examples in terms of properties related with the target concepts. Secondly; decision tree method is very effective since it requires a general calculation with regard to a number examples that are generally observed, and finally a decision tree creates a representative concept, meaning of which can be easily explained. A decision tree consists of four principal components as decision nodes, chance nodes, possibilities and returns. Decision nodes are depicted with boxes and define the special decision that the decision maker faces with. Each branch of the decision node is related with the possible decision and chance node which is depicted with circles in diagrams indicate the certain game that the decision maker faces with. Each branch of the chance node is related with the potential outcome of the game and each outcome has a possibility. Sum of the possibilities pertaining to the outcomes of change nodes is equal to 1. Returns are indicated on the tip of each branch located on the right of the decision tree. Return is the value of each possible combination and risky outcomes of the options. If the decision maker is risk-neutral the returns are monetary value, if the decision makers are risk-averse or risk-seeking the returns are utilities related with value. Let’s assume that a cement manufacturer finds a new market abroad and therefore, the company is to make a decision to own a plant with three different capacities as large, medium and small. In this example the capacity and size of the plant that the company will own will directly depend on the demand amount. • If the demand is high and the company a) Owns a large plant it will realize a return of TL 105 million, b) Owns a medium plant it will realize a return of TL 90 million, c) Owns a small plant it will realize a return of TL 75 million. • If the demand is at medium level and the company a) Owns a large plant it will realize a return of TL 50 million, b) Owns a medium plant it will realize a return of TL 55 million, c) Owns a small plant it will realize a return of TL 50 million. • If the demand is low and the company a) owns a large plant it will realize a return of TL 20 million, b) Owns a medium plant it will realize a return of TL 30 million, c) Owns a small plant it will realize a return of TL 45 million. In this example, the company would have made a large plant investment and maximize its return if it definitely knew that the demand would be high. Similarly, the company would have preferred to have a small plant if it knew that the demand would be definitely low. However, demand of cement is uncertain in this case. Let’s assume that the company believes that the demand will be realized at high, medium and low levels with the possibilities of 40, 20 and 40%, respectively. Assuming that the company is risk-neutral, the returns will be monetary values and will indicate the net return of the company for each option.
Decision node A splits the choice of the company in three different options as large, medium and small. Decision nodes B, C and D represent the games that the company will encounter according to the decision in node A and each game has three different outcomes. These outcomes are high, medium and low levels of realization of the demand and their possibilities are of 0.4, 0.2 and 0.4, respectively. The return of the company depends on the decision it will make in node A and the actual outcome of the game in relation to that. If the company decides to have a large capacity plant in node A it will realize a return of TL 105, 50 or 20 Million depending on the realized demand being high, medium or low, respectively. In the case that the company decides to have a medium size plant in node A it will realize a return of TL 90, 55 or 30 Million depending on the realized demand being high, medium or low, respectively and if the company decides to build or buy a small capacity plant in node A it will have a return of TL 75, 50 or 45 Million depending on the realized demand being high, medium or low, respectively. The company will calculate the expected value of each game and choose the game which has high expected value in node A in order to choose the optimal action plan. Therefore, the company assesses the tree from right to left and the expected values of the game in chance nodes B, C and D can be calculated as follows;
Thus, expected value of the game in chance nodes B, C and D are calculated as TL 60, 59 and 58 Million. In this case, the optimal decision is to have a high capacity plant which has a high expected value (TL 60 Million). However, real strength of the decision trees and their usefulness become apparent in cases where multiple decisions are evaluated and analyzed in a sequential way. Let’s consider that a partnership is proposed to the manufacturer company for a medium size plant in this example and the company would make a decision either to accept it or not. In order to solve a problem regarding sequential decisions it is required to set off from the terminal nodes and move backwards and also to take the most appropriate action to continue with each step. In the current example the company primarily needs to calculate what it will do in node B in order to know what it can do in node A. This method is called as “backward induction” and allows to separate the problem into smaller pieces for easier resolution in case of a complex sequential decision. Whereas, the company expects to achieve a return of TL 60, 67.5, and 58 million by owning a large, medium or small size plant. The company will opt for buying a medium size plant which has a higher expected value. However, the company may realize the expected return of TL 67.5 million by buying the medium scale plant only if the partnership offer is accepted. Therefore, the company shall primarily decide if it will accept the partnership offer. Hence, if the company refuses the partnership offer, it can realize an expected return of TL 59 Million from this medium size investment. In this case, the decision of a company which would like to maximize its expected returns would be investing in a medium size plant by accepting the partnership offer. Expected Utility and Von Neumann-Morgenstern Utility Function Decision trees analyzed in the above sections demonstrate the various options clearly and may establish a connection between the actions, however, may fail to assess such outcomes fully. The reason for that is the expression of outcomes of all alternatives in the decision trees in the above examples in terms of expected value. Utility function expected in the solution of the problem can be featured in case of certainty, yet, the same solution and method can be employed also in case of uncertainty, therefore, outcomes of all alternatives in the decision trees can be expressed in expected utility instead of expected value. Although expected value and expected utility functions seem to be similar they are distinctly different from each other. Value function defines the attitude of the decision maker towards the risk and utility function defines the attitude of the decision maker towards both the asset and the risk. While all utility functions are also a value function, value functions may not be a utility function. Utility function is equal to value function only in the case that the decision maker is risk-neutral. Since calculation of the expected value is not always sufficient to analyze the decisions made in cases of uncertainty, John Von Neumann and Oskar Morgenstern (vN-M) proposed a standard utility maximization model which also includes the risks. Therefore, the primary analytical approach in terms of uncertainty is based on the basics of the study of Von Neumann and Morgenstern. Since consumer behavior is very important in case of risky situations one of the most useful features of vN-M utility theory can be expressed as clear modeling of the risk by the theory. Hence, vN-M utility theory was structured on the decision maker’s choice for the conversion function considering the attitude towards the preference between risky and non-risky situation. In the formula regenerated by the researchers, a rationalist individual wants to maximize the expected utility. However, variables of the utility function will be dependent variables in this case. Let’s consider that number of possible circumstances is equal to n, outcome dependent on the first circumstance in case of uncertainty is equal to x1 and realization possibility of such outcome is equal to p1 . If outcomes which correspond to each possible circumstance xn, and the related possibilities are to be defined in the same way until the expression of pn utility function which demonstrates all possible outcomes can be expressed as U(x1 , x2 ,…. xn ; p1 , p2 ,…. pn). Changes in the possibilities included in this function may change the choice of the decision maker and possibilities of various circumstances can be included in the utility function in a more complex manner from the theoretical aspect. Therefore, when utility in each case is calculated and weighted the following formula is obtained; This form of the utility function, which represents the indifference curves of a consumer for risky games is known as Von Neumann-Morgenstern (vN-M) utility function. Utility of a game in vN-M utility function is the expected value of the utility of each outcome. Expected utility is the weighted average of the utilities achieved from each possible return and can be expressed with the following formula; Whereas; EU: expected utility pi : i. possibility of outcome U(xi ): i. utility of outcome. vN-M As previously defined, in utility function method, goal of an individual shall be to maximize the expected utility. In other words, when vN-M utility function is used in decision making process, the critical point is that the individual or the company aims to maximize the expected value of the utility rather than the utility of the monetary value. This can be explained by restudying the example, the decision tree of which is illustrated in Figure 1. In this example, the company has decided to own a large, medium or small capacity plant and then decided with the option of owning a large plant with the highest expected value. Let’s assume that return dependent vN-M utility function of the cement manufacturer is U=√R. In this case, the company will make its decision by considering capacity sizing of the plant they will own and prefer the option which offers the highest expected utility. Figure 3 shows the decision tree which includes return dependent utility values of the cement manufacturer. Here, the expected utilities of the game in chance nodes B, C and D can be calculated as follows; In this case, the company choose option D which offers a utility of 7.56 units since the higher one of the expected utilities will be preferred rather than the expected return. In other words, the company will want to own a small plant. Expected utility function has very suitable characteristics for the analysis of choices to be made under uncertainty. For instance, let’s assume that a cement manufacturer experiences an uncertain situation whether to make a new investment or not. Considering that the capital of a manufacturer which has a capital of TL 20 Million reserved for this task will increase-decrease by TL 5 Million depending on the outcomes of this investment, the capital of the manufacturer will be TL 15 million with a probability of 50% and TL 25 million with a probability of 50%. However, these figures express the expected values. Yet, expected utility concerns the attitude of the manufacturer against risk unlike the expected value. In this example, expected value of the investment is TL 20 Million (15×0,5+25×0,5). However, a riskaverse investor will not prefer to make such investment, a risk-neutral investment will be indifferent whether to make the investment or not and a risk-seeking investment may prefer to make the investment.