The decision-making process of individuals in a competitive environment under risk and uncertainty - Менеджмент и трудовые отношения курсовая работа

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The decision-making process of individuals in a competitive environment under risk and uncertainty

Value and probability weighting function. Tournament games as special settings for a competition between individuals. Model: competitive environment, application of prospect theory. Experiment: design, conducting. Analysis of experiment results.


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In modern world with its free markets and globalization, competition becomes more and more common environment. The world population continues to grow, and toughness of competition increases. Rating systems develop in different areas such as education, and best specialists such as salesmen in business receive bonuses etc. Some companies implement competition features, which are thought to increase the productivity of workers .Competition is also a key element in such areas as sports and politics.
However, outcomes in spreading competition do not fully depend on actions of people only. Some random factors may influence the results no matter how much effort has been put into performed actions. In this case attitude of people towards risk influences their actions and the result of the whole competition. Therefore, it becomes important how risk attitudes may change in the competitive environment. In this research we analyze behavior of individuals in the competitive environment under risk and uncertainty. We apply prospect theory (D. Kahneman, A. Tversky, 1979) to this setting, which “has become one of the most influential behavioral theories of choice in the wider social sciences, particularly in psychology and economics” [14]. We construct a model based on prospect theory application to the competitive environment, when wealth is not directly involved in the competition itself. In this case applicability of prospect theory and presence of its effects requires experimental verification. We conduct an experiment for decision-making process under risk and uncertainty in the competitive environment. Then we analyze results of the experiment and test the compliance between these results and theoretical model based on the prospect theory. As a result, we can evaluate the applicability of prospect theory to the competition and realization of such effects as reference dependence and reflection effect in the considered setting.
Goals and objectives of the research
Goal of the research: to evaluate the decision-making process of individuals in a competitive environment under risk and uncertainty by applying prospect theory in the research model and conducting an experiment.
Consider theoretical aspects of the decision-making process of individuals under risk and uncertainty.
Apply prospect theory to the competitive environment with uncertainty elements and construct a relevant model.
Develop and conduct an experiment to test an application and effects of prospect theory to the competitive environment with uncertainty elements.
Analyze the results, which reflect behavior of individuals in the decision-making process of individuals in a competitive environment under risk and uncertainty.
Decision-making process is a process of rational or irrational choice between alternatives, aimed at achieving some result. Outcomes are often uncertain, and people face the risk in the process of decision-making. Such process becomes more difficult under risk and uncertainty, and many researches study this case.
The expected utility hypothesis takes a central place in studies of decision theory and is based on the assumption of the rationality of economic agents. In general, the purpose of the agent is to maximize expected utility: У (pi u(xi)), where xi - value of the outcome, pi - probability of its implementation, and u(xi) is a utility function dependent on the outcome. This theory was developed as an answer to so-called St. Petersburg paradox. This paradox is a setting, when agents must be ready to pay an infinite amount of money for a certain lottery, if they base a decision on expected value. In reality, it is not true, and expected utility concept can explain why. However, there are situations where an individual's behavior is not consistent with the hypothesis of expected utility too.
In the traditional economics, one crucial property of economic agents is assumed. People are rational; they analyze and take into account all available information when making decisions. However, behavioral economics based on the evidence from real world proves that people are irrational; their choice is largely intuitive, and there are cognitive heuristics; people are sensitive to many parameters in decision-making; risk is taken or avoided depending on the context.
Various theories arise in order to describe real behavior of economic agents and deal with phenomena, which are hardly explained by existing decision-making theories. Much credit for the development of alternative economic concepts of decision-making belongs to Daniel Kahneman, who has received the Nobel Prize in Economics in 2002 “for having integrated insights from psychological research into economic science, especially concerning human judgment and decision-making under uncertainty”.
In 1979 Daniel Kahneman and Amos Tversky proposed a so-called “prospect theory” [5]. This theory is based on the real behavior of economic agents and can be applied to various settings. We consider prospect theory in details in order to apply it in our research.
Agent have a value function of a particular form. This form is based upon existence of various effects, and function itself is used in the process of decision-making. Description of these effects and conclusions about agents' behavior are presented below.
Outcomes are considered in terms of gains/losses, where gains/losses are deviations from some reference point. This is a reference dependence effect. In standard case, reference point is an initial wealth of given agent. Thus, agent stays in the status quo position, when nothing is gain or lost.
Reflection effect signifies that value function is concave for gains and convex for losses. Loss aversion suggests that for same sizes of gain and loss, loss affects the valuation more. It means the value function if not symmetric, and it is skewed for the negative domain. Diminishing sensitivity is an effect, which reflects that each additional unit in either gain or loss affects the valuation by less than previous unit.
As a result, agents' value functions have asymmetric S-shapes (See Figure 1).
Figure 1 - Prospect theory: value function (Kahneman, Daniel, and Amos Tversky (1979) “Prospect Theory: An Analysis of Decision under Risk”, Econometrica, XVLII, 263--291)
Prospect theory suggests that decision process consists of two stages: editing and evaluation. On first stage (editing), an individual orders outcomes of a decision according to certain heuristic. In particular, people set reference point and classify outcomes as gains or losses. On second stage (evaluation), individual associate each decision with certain value of utility.
Value function v(z) depends on the outcome z, and for each decision there is a set possible values of this function. Utility function u equals to weighted sum of values of possible outcomes. Prospect theory and its improved version (cumulative prospect theory) suggests the non-linear probability weighting [6].
Evidence indicates that decision-makers have such probability weighting function w, that they overweight small probabilities and underweight large probabilities (See Figure 2).
Figure 2 - Prospect theory: probability weighting function (Kahneman, Daniel, and Amos Tversky (1992). "Advances in prospect theory: Cumulative representation of uncertainty". Journal of Risk and Uncertainty 5 (4): 297-323)
Individual i calculates utility for each decision with outcomes zj and probabilities pj according to functions w(p) and v(z).
Prospect theory suggest that this individual then makes a decision with maximum utility associated with it. In this case introduction of probability weighting function is an important distinction from other theories, and this function can also be called an effect of prospect theory.
Effects suggested by prospect theory are powerful enough for explaining particular cases with behavior, which does not agree with existing theories. For instance, Colin Camerer considers such examples in "Prospect Theory in the Wild: Evidence from the Field" [2]. This research provides ten patterns of observed behavior, which can be considered as anomalous for expected utility theory. However, such behavior can be explained by just three components of prospect theory. It shows the advantage of prospect theory when dealing with real behavior.
In various researches prospect theory is applied to different contexts. In particular, we can be interested in application to political science (Druckman 2001; Lau and Redlawsk 2001; McDermott 2004; Mercer, 2005; Quattrone and Tversky 1988) due to the fact that politics is a completive environment. In particular, prospect theory is applied to the areas of international relations (Berejikian 1997, 2002; Faber 1990; Jervis 1994, 2004; Levy 1994, 1997; McDermott 1998), international political economy (Elms 2004), comparative politics (Weyland 1996, 1998), American politics (Patty 2006), and public policy (McDaniel and Sistrunk 1991).
Prospect theory is not the only theory, which is based on real behavior of people and can explain various irrational decisions. Alternative theories include theories based on heuristics analysis and bounded rationality concept, developed by Herbert A. Simon in 1956. Heuristics cause biases and so-called cognitive illusions [13, 3]. Over-confidence or false self-confidence can be caused by optimistic bias, illusion of control, expert judjment, or hindsight bias. Researches by Swenson O (1981) or McCormick, Iain A., Frank H. Walkey (1986) show distortions caused by optimistic bias [18, 11]. In addition, it is shown that for random outcomes players act as if they can control these outcomes, while taking more risk [4, 16].
In our research we consider prospect theory as an advanced theory suitable for different settings. We apply it to the competitive environment, where uncertainty can be an unavoidable feature due exposure to a huge variety of factors of different nature including actions of competitors.
game individual environment experiment
Tournament games are special settings for a competition between individuals. Systems of bonuses for best sellers, sports and other real-world contexts are described with help of tournaments games. Researches include different types of games such as auctions.
In most games, wealth is directly involved (e. g. in form of effort), however prospect theory is still rarely used in tournament games. In addition, there is also case of so-called winner-gets-all condition of the game. In this case only best player gets the prize. Application of prospect theory to such competitive environment is not straightforward due to the fact that in dynamic game wealth is not involved directly.
We apply prospect theory to the case of dynamic winner-gets-all tournament game, which is not done by other researches yet. Application of our research can allow deep analysis of real-world situations with help of one of the most advanced theories in the area of decision-making.
In the previous section, we looked at various theories that can be applied to the competitive environment with uncertainty element. Primarily, we wish to analyze application of prospect theory to tournament games. This section is used to present a particular model for our research. Following model is constructed for two players who compete in a game with predetermined rules. We believe that such model can be used as foundation for further researches with more people involved.
Consider a game between two players with following rules. Game is dynamic and consists of several identical rounds. In each round both players make economic decisions with a chance of gaining points in the game. Both players start with zero number of points, and points are accumulated throughout the whole game, so players have their scores. In the end of the game (after last round) player with higher score wins the game and gets real-world prize (e.g. money). Another player gets nothing. Also each player receives half of the prize, if scores are equal. This setting is a case of winner-takes-all game type from tournament games, and we can move on to detailed description of the model.
Each round of the game consists of three stages.
Players simultaneously make choices over a set of risky lotteries that may yield points.
Players determine how many points they actually gain from lotteries chosen.
Players are given a following set of n lotteries:
, k = 1..n, where ak > 0 and ak increases with k
Potentially there can be a continuous case, when player can possibly get any number of points between a1 (with 100% probability) and an, so n > ? with fixed an.
For each lottery Ak we introduce notation for probability pk = a1/ak - probability of receiving ak points in lottery Ak.
All these lotteries have equal expected values and different variances, which increase with possible gain:
Choice over lotteries reflects risk attitude of the player. For instance, choice of riskless lottery A1 shows risk aversion of the player in the respective round. Player's preferences over lotteries may change during the game, and we need to construct a model to explain the process of making choices using economic theories.
At first, consider decision-making process in first and last rounds of the game with many rounds (for example, with 40 rounds). In first round, game is too far from its end and winner determination. Players are targeting accumulation of points, as it is a way of getting the prize in the future. This process will be analyzed in details further in this section of the research. In contrast, strategy in last round is different, because outcome is purely expressed in terms of prize. In addition, there can be some end-game effects like big disadvantage causing some lotteries to be useless. So by the end of the game its structure becomes closer to standard games with payoffs of prizes and not points. Decision-making relies less upon points themselves and their accumulation. Length of the game directly influences weight of end-game effects, and in a long game there are more rounds, when points are key target.
The research is focused on the process of accumulating points, when players are not yet concerned about prize itself. End-game effects are not considered, and all game rounds are seen as identical in terms of motivation - accumulating points. Players wish to increase the overall chance of getting prize by acquiring maximum possible advantage during the game. Therefore, the question is how exactly points are incorporated into decision-making process in each round of the game.
We refer to participating players as “player 1” and “player 2” together with following denotations for a chosen round (for example, round t):
yi - score of player i at the beginning of the round, which is number of points accumulated by player i before current round.
xi = yi - yj , where i ? j, i = {1,2}, j = {1,2}. Variable xi reflects the difference between players, and we refer to it as to an advantage of player i. This advantage can be negative.
yi* - score of player i at the end of the round.
xi* = yi* - yj* , where i ? j, i = {1,2}, j = {1,2}. This is an advantage of player i in the end of the round, thus xi* in round t becomes xi in round t+1.
Score of player does not matter on its own, and higher advantage leads to higher probability of winning the game in the future.
For the whole game utility function of a player positively depends on the point advantage. We apply prospect theory framework to players' utility functions. Improved version of prospect theory - cumulative prospect theory - is used further in the research.
In particular, we are interested in the reference dependence effect. Individuals are concerned about deviations from some reference point - gains and losses. Application of reference dependence determines how to consider other effects such as loss aversion, reflection effect, and diminishing sensitivity.
Therefore, we need to understand how reference point is determined in a competition with accumulating points. The problem is that we cannot take the notion of initial wealth and apply it to the game. Wealth does not change throughout the whole game, and only points can be earned in the process. Prize can change the wealth level, but it is the only and delayed way. Therefore, we need to come up with interpretation of the concept of reference point for our special setting.
We consider a situation, where both players begin new round of the game with equal scores, which means that xi = 0 for i = {1,2}. Players care about deviations from this situation, and they consider such deviations as gains/losses. Reference point for player i will be xi = 0 and this is status quo for both players.
Now we can also apply other effects of prospect theory, when reference point is determined. Thus loss aversion, reflection effect, and diminishing sensitivity are also incorporated to the model.
Suppose players have value functions vi, i = {1,2}. These functions positively depend on the advantage and satisfy prospect theory, as described above. For each player point A0 reflects value of the function vi for advantage xi in the beginning of the round. When advantage becomes xi*, player gets different value of the function, if xi ? xi*.
Alternative interpretation is that reference point of the player moves together with rival's score. In this case value function depends on player's score and shifts when rival's score changes.
Figure 3 illustrates both interpretations with A0 for positive xi. Note that we can base it around player 1 due to symmetry of the game. Reflection point is x1 = 0 (a) or y1 = y2 (b), which is equivalent to x1 = 0. Depending on the interpretation, different value functions need to be used for the same player 1 (v1a and v1b). Further in the research, we use first interpretation, so utility function depends on the advantage, and v1 is equivalent to v1a.
Figure 3 - Application of prospect theory to the competition between two players
After applying prospect theory to a tournament game, we move to the analysis of decision-making process for considered setting. Rules are identical for all rounds, and rounds are linked with each other through global parameters (score). In the beginning of each round players have full information about all previous game results including choices made by both players and changes in scores. Players make choices independently in each round, and the only inputs for decision-making process and value functions are current scores yi and advantages xi derived from them. In this case, analysis of actions for one round is the same as analysis of a static game (one round). Results will be then used for all rounds, when players wish to accumulate points (no end-game effects).
Concept of Markov property can be employed for such claim [10]. Markov assumption suggests that considered dynamic process has a memoryless property. So value of function vi in any round is only influenced by the parameters of the round that directly preceded it. Among all parameters from other rounds, player uses only resulting advantage of the previous round, which we call initial advantage xi in current round of the game.
In the beginning of each round, players choose one of the lotteries Ak based on scores yi according to their utility functions with prospect theory effects. In our model, we also apply cumulative prospect theory, and value function is included into the utility function with addition of probability weighting function. Prospect theory suggests that individuals overweight small probabilities and underweight moderate and high probabilities.
Suppose players have value functions vi and probability weighting functions wi. Then prospect theory suggests that overall utility ui of player i from the lottery with outcomes zj and corresponding probabilities pj, j =1..m, is calculated in the following way:
In each round there are several possible values of the resulting advantage xi*. This advantage is a change in initial advantage xi due to possible gains of both players, which are subject to uncertainty.
Both players make their choices simultaneously, and game theory matrix can reflect outcomes of players' interactions. Set of lotteries is reduced to following two lotteries (n=2), so we can then draw an important conclusion from matrix analysis:
Both players can choose either A1 or A2. New advantage xi* consists of current advantage xi changed by possible gains from lotteries. Scores may change by ?yi = yi* - yi, so ?yi is an outcome in a chosen lottery. Following is true:
Matrix for xi* is a zero-sum game, and we right down such matrix for x1*.
Figure 4 - Matrix for advantage xi* for two lotteries
We know that advantage xi* is an argument for value function vi of player i. Let us consider an evaluation by player 1. This player determines values of function v1 for each possible increase in rival's score ?y2. Player starts round in point A0, where an advantage is x1. For some value of ?y2 player can potentially move either of two points, if lottery Ak is chosen.
First, player can move to point Ak* with probability pk, so ak points are gained:
Second, player can move to point A0* with probability (1-pk), so no points are gained:
Figure 5 illustrates changes in the advantage for two lotteries and some value of ?y2. Depending on the lottery chosen (A1 or A2) there can be three different outcomes (A0*, A1*, A2*), while round starts in A0. Notice, that if player chooses riskless lottery A1, then there is always movement from point A0 to point A1*.
Figure 5 - Values of function v1(x1*) for given value of ?y2
If we go back to the matrix of xi*, then there is a corresponding value of ui for each set of potential xi* values and vi(xi*) values. We can form a matrix for values of utility function using cumulative prospect theory.
Such matrix is not a zero-sum game due to differences in individual functions vi and wi, though x1* = -x2* and x1 = -x2. Each v1(x1*) corresponds to v2(-x1*), and we construct matrix for payoffs of player 1 only. Also recall that p2 = a1/a2 and wi(1) = 1.
Figure 6 - Matrix for overall utility of player 1 (function u1) for two lotteries
Assuming choices in different rounds are independent, evaluation of lotteries for player i in a given round depends on the following:
set of lotteries Ak - variables a1 and a2;
value function vi and probability weighting function wi;
Only xi changes throughout the game, and individual functions stay the same together with given lotteries. However, we cannot solve the matrix for utility and predict decisions made by players without knowing individual functions wi and vi.
Let us show that in the same game (fixed set of lotteries) different people have different preferences over lotteries for the same advantage. Consider a following example with player 1. Further in the research we refer to it as to “Example #1”.
Set of lotteries includes lotteries A1 and A2, and a1 = 1; a2 = 2. Thus p2 = a1/a2 = 0.5. Initial advantage in the round is x1 = 2. For non-negative values of advantage player 1 has value function v1(x1*) = (x1*)0.5. Player overweights small probabilities as it is predicted by the prospect theory, and probability weighting function w1 take following values:
Now we can calculate values of utility function u1 for each combination of lotteries chosen by both players (See Figures 6 and 7).
Figure 7 - Matrix for overall utility function u1 for two lotteries in the Example #1
Let us calculate approximate values of function ui.
Figure 8 - Matrix for approximate values of u1 in the Example #1
In a given game for a particular individual functions vi and wi player 1 wish to choose same strategies as player 2, when x1 = 2. Values of utility function relate as follows: 1.41 > 1.37 and 1.45 > 1.37.
Now consider another example - Example #2.
Players 3 and 4 play a game with same lotteries as in Example #1. In current round of the game player 3 also has same advantage as player 1: x3 = x1 = 2. Value functions of players 1 and 3 are the same too: v1(x1*) = (x1*)0.5 and v3(x3*) = (x3*)0.5. However, players 1 and 3 have different probability weighting functions, and this is the only difference. Player 3 simply overweights small probability less than player 1:
Let us go straight to the matrix for approximate values of u3.
Figure 9 - Matrix for approximate values of u3 in the Example #2
In contrast with player 1, player 3 has a dominant strategy - choosing lottery A1 in this round of the game. This yields higher values of utility function u3 for both possible actions of player 4: 1.41 > 1.37 and 1.37 > 1.30.
Considered examples show that individual functions matter. People with different value function also may have different preferences over lotteries other things being equal. This creates a problem for solving the model.
Now we can go back to the case with n lotteries and summarize our model and its solution. Notice that for any number n, matrixes with xi* and ui (See Figures 4 and 6) can be easily expanded. We consider a dynamic tournament game with winner-takes-all condition and necessity in accumulating points by choosing risky lotteries. In this setting, we apply prospect theory and construct a model for decision-making process in each round of the game. We conclude that application of prospect theory allows us to analyze changes in risk attitude for a case of tournament games. However, solution for constructed model depends on preferences of particular players. Thus, next step in the research is conducting an appropriate experiment (Section 3 of the research) with real players. In section 4 we analyze collected data and test applicability of prospect theory, which is described by our model.
In our research, we design and conduct an experiment to study behavior of individuals in the competitive environment with uncertainty elements. Design of the experiment is based on the model of the research described in section 2.
We present a dynamic game with two participating players. In each game players compete for a prize in a series of consequent rounds. The prize is a certain sum of money, which is same and known for each game. There are no costs for participants except that they spend time and effort during the game.
Players can gain points in each round, and points are accumulated for each player forming their scores. If player gains some number of points, then score of this player increases by this number. Initially, both players have scores of zero points.
An important element of the game design is how game ends. All rounds of the game are identical in a sense of rules, and game consists of several rounds of the game. We believe that obtaining information for 40 rounds in each game is enough for data analysis with reliable results. However, as described in section 2, we expect that there will be distortions due to end-game effects for the game with known number of rounds. Players are expected to concentrate less on points and change their behavior in order to obtain the prize. We wish to purify experiment results from these distortions by using special rules for the game end.
Players do not know after which round game will end. We tell them that game ends after round with predetermined number known by the researcher only. Such number is 40. We write down this number and reveal it only after players have finished 40 rounds. Players can see that this number is predetermined and written down, so the game is fair, and researcher cannot end the game in order to help one player.
With such game design, we make sure that there are no end-game effects, and players are concentrated on earning points. Game can end at any time, therefore in each round scores are crucial. Having advantage/disadvantage becomes the primal concern, and such game design suits the research model.
Rules for rounds of the game are very close to those in the research model. In the beginning of the round, each player chooses one of four available lotteries (“A”, “B”, “C”, and “D”) with following parameters:
These lotteries satisfy the model and its set of lotteries of Ak completely (with n = 4). Players choose lotteries simultaneously. Choices are made independently, and player cannot observe rival's action until choices are made by both players. Then players determine how many points they actually gain and how scores change. Next round begins.
For described game, we use a special form (See Attachments). Before the game begins, each player receives printed copy of such form, pen and a dice with 6 sides. We use standard dices with 1 to 6 dots on each side. Description for elements of the form follows below.
Before the game begins, each participant gets a number associated with this player only. When two players are paired for a game, each player needs to write down two numbers in a relevant space of the form: own number and rival's number (See Figure 10).
Next element of the form is a description of 4 available lotteries. This description is designed in such a way, that players can easily access it at any time. Figure 10 illustrates first part of the form - two elements described above.
Figure 10 - First part of the form used for an experiment
Second and last part of the form allows players to write down what happens in the rounds of the game (game log). Form is designed for 59 rounds, and it also includes round number zero. Information for one round consists of exactly 7 cells in a row separated by space as 5 and 2 (See Figure 11). First cell corresponds to the number of round (“#”). Instead of having a long list of rounds, we have 2 large columns. Rounds 0-29 are located in the first column, rounds 30-59 - in the second column. Players start in the first column with round 1 and play round by round going over to th
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