The decisionmaking 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.
Рубрика  Менеджмент и трудовые отношения 
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Язык  английский 
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Introduction
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 decisionmaking 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 decisionmaking process of individuals in a competitive environment under risk and uncertainty by applying prospect theory in the research model and conducting an experiment.
Objectives:
Consider theoretical aspects of the decisionmaking 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 decisionmaking process of individuals in a competitive environment under risk and uncertainty.
Section 1. Literature review
1.1 Choice under uncertainty
Decisionmaking 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 decisionmaking. 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 socalled 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 decisionmaking; 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 decisionmaking theories. Much credit for the development of alternative economic concepts of decisionmaking 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 decisionmaking under uncertainty”.
In 1979 Daniel Kahneman and Amos Tversky proposed a socalled “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 decisionmaking. 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 Sshapes (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, 263291)
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 nonlinear probability weighting [6].
Evidence indicates that decisionmakers 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): 297323)
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 socalled cognitive illusions [13, 3]. Overconfidence or false selfconfidence 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.
1.2 Tournament games
game individual environment experiment
Tournament games are special settings for a competition between individuals. Systems of bonuses for best sellers, sports and other realworld 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 socalled winnergetsall 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 winnergetsall tournament game, which is not done by other researches yet. Application of our research can allow deep analysis of realworld situations with help of one of the most advanced theories in the area of decisionmaking.
Section 2. Model
2.1 Competitive environment
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 realworld 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 winnertakesall 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 update their scores.
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 decisionmaking 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 endgame 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. Decisionmaking relies less upon points themselves and their accumulation. Length of the game directly influences weight of endgame 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. Endgame 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 decisionmaking 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.
2.2 Application of prospect theory
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 decisionmaking 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 decisionmaking 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 endgame 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:
, , 0 < a1 < a2
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 zerosum game, and we right down such matrix for x1*.
Player 2 

A1 
A2 

p1 = 1 
p2 = a1/a2 
1  p2 = 1  a1/a2 

+ a1 
+ a2 
0 

Player 1 
A1 
p1 = 1 
+ a1 
x1 
x1+a1a2 
x1+a1 

A2 
p2 = a1/a2 
+ a2 
x1+a2a1 
x1 
x1+a2 

1  p2 = 1  a1/a2 
0 
x1a1 
x1a2 
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 (1pk), 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 zerosum 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.
Player 2 

A1 
A2 

Player 1 
A1 

A2 

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:
initial advantage xi;
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”.
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 nonnegative 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).
Player 2 

A1 
A2 

Player 1 
A1 

A2 

Figure 7  Matrix for overall utility function u1 for two lotteries in the Example #1
Let us calculate approximate values of function ui.
Player 2 

A1 
A2 

Player 1 
A1 
1.41 
1.37 

A2 
1.37 
1.45 

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.
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.
Player 4 

A1 
A2 

Player 3 
A1 
1.41 
1.37 

A2 
1.37 
1.30 

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 winnertakesall condition and necessity in accumulating points by choosing risky lotteries. In this setting, we apply prospect theory and construct a model for decisionmaking 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.
Section 3. Experiment
3.1 Design of the experiment
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 endgame 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 endgame 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 029 are located in the first column, rounds 3059  in the second column. Players start in the first column with round 1 and play round by round going over to the second column, when they get to round 30. Figure 11 illustrates beginning of the game log and first 6 rows.
Figure 11  First 6 rows of the game log in the form used for an experiment
We describe how players use game log with an example of round 1.
For the whole game players sit at the table against each other. When round 1 begins, each player does the following:
Covers columns 2 to 5 in the line corresponding to round 1 with one hand so rival cannot see them. These 4 cells are located in the third line (round 1) in columns 2 to 5 (“A”, “B”, “C”, and “D”).
Chooses a lottery and make a mark (e.g. a check) by a free hand in the corresponding cell, which is still covered. Player chooses one lottery, so there must be exactly one mark for each round.
Waits until both players will make decisions, still covering 4 cells with a hand.
Reveals what lottery is chosen.
Rolls a dice.
Determines how many points lottery actually yield. Description of lotteries includes numbers players need to roll in order to gain points in a chosen lottery.
Updates the score. Score increases by a lottery gain, if roll is successful or lottery A is chosen.
Writes down the result in column 6.
Informs rival about the updated score.
Writes down rival's score in column 7.
Described process is repeated in each round. When players reach round 41, researcher stops the game and shows players the paper with number 40 written down. It justifies that game is stopped in the right, predetermined moment. Then player with higher score is awarded with a prize. Player with lower score gets nothing. If scores are the same, prize is divided between players.
The most important step is a step number 2, when players simultaneously make decisions. They may look on the information about previous rounds (filled game log) and primarily on the scores for the beginning of this current round. When round t begins, players have scores written down in columns 6 and 7 in previous line (round t1). They can easily see what advantage/disadvantage they have. Based on this information, players make their choices in such competitive environment under risk and uncertainty.
For each player, experiment yields values of two necessary variables for all of 40 rounds. These variables for player 1 in a pair are (and for player 2 symmetrically):
x  initial advantage, which is a difference between score of player 1 and score of player 2 in the beginning of the round of the game;
a  gain of the chosen lottery (1, 2, 3, or 6) in the round of the game, which shows what level of risk is optimal for the player.
Therefore, we know how each player reacts (variable a) to the advantage/disadvantage (variable x) in the beginning of the round.
3.2 Conducting the experiment
An experiment was carried out between 14.06.2013  17.06.2013 in Moscow, Russia.
In the experiment there were 30 participants in the age group 1723 years, including 16 women and 14 men. Average age was 19.3 years. All participants were either undergraduates (26) or graduates (4).
For the experiment we appealed to visitors of special coffeehouses and clubs for people interested in intellectual games (e.g. chess). These people have experience of strategic interaction with rivals in the games and know what competition is. This fact together with small age variance and occupation allow us to talk about relatively homogenous perception of all participants.
Each participant played one game with previously unknown person of same gender. In total, 15 games were played with average length of 15 minutes. During the game players were supervised, so there was no interaction with other people in the room. We conducted up to three games at the same time. Choice of place for an experiment also provided quiet and calm environment.
Before the game, each player received printed copy of the form used for an experiment, pen and a dice with 6 sides. Experiment was carried out in Russian language, thus we used translated version of the form (See Attachments). Prize was chosen to be 200 Russian rubles. In every game there was a winner, so prize was never divided between two players.
Results of all 15 games with 40 rounds each were then transferred to the electronic form for further analysis in a specialized software.
Statistical analysis of the results of the research is carried out by using such software as Stata 12 and program's features of Microsoft Excel 2013.
Statistical hypothesis testing is performed using one sample median test (Wilcoxon signedrank test). Ordinary least squares method is used in order to estimate unknown parameters in a linear regression model for risk attitude changes.
Section 4. Analysis of experimental results
Experiment described in previous section provides us data for all games played. This includes choices made by players, outcomes of chosen lotteries, and score changes. For the model, we assume that decisionmaking process is independent in each round of the game. Also it is assumed that player bases a decision on player's advantage in the beginning of the round, and no information from other rounds is used.
Therefore, from all available data we only need data on initial advantage and player's choice in each round of the game. Variable x corresponds to the initial advantage, and variable a corresponds to possible gain of a chosen lottery. So for each player we know the reaction (variable a) to the advantage/disadvantage (variable x) in the beginning of each round of the game (See Table 1 in Attachments).
In section 2 we show that testing applicability of prospect theory to competitive environment requires one crucial thing. This thing is knowing individual functions, such as value function and probability weighting function, for each player.
There are 4 discrete alternatives in each round of the considered game. Research model suggests that utility from the lottery depends both on its parameters and on the attributes of the player. Analysis of such case can be done using conditional logit model. McFadden (1973) proposed modeling the expected utilities in terms of characteristics of the alternatives rather than attributes of the individuals [12]. Such attributes are unknown so the model becomes very useful, when players choose one lottery given set of 4 discrete available lotteries with certain parameters (ak, pk). However, independence of irrelevant alternatives (IIA) must hold for such model. This concept suggests that individual's preference between two alternatives is not affected by the introduction of a third alternative.
However, Kahneman and Tversky (1982) suggest that real behavior of people violates the IIA [7]. Therefore, using conditional logit for testing prospect theory may yield unsatisfying results. In the process of data analysis we have tried to use conditional logit model, and results are not indeed useful as it is predicted. Thus, we need to apply another method for testing application of prospect theory to competitive environment.
Application of prospect theory can provide us with certain patterns of players' behavior. These patterns are not precise predicted preferences over lotteries in each round. On the other side, testing the existence of such patterns does not require knowing individual functions of players. These patterns are more general, and we need to derive them first and then test their realization on our data. Possible evidence from the data creates a signal that prospect theory is applicable to competitive environment according to the constructed model.
Let us consider the case, when player has big initial advantage xi. Suppose that with such big advantage player will still have positive advantage for all possible outcomes xi*. It means that in a given round player always remains with lotteries Ak on the positive domain of value function with risk aversion property (See Figure 12). Thus, with such risk attitude player chooses small potential gain with no risk  lottery A1.
Figure 12  Case of big advantage and risk averse behavior of player i.
Similarly to the previous case, in the case of big disadvantage player remains in area corresponding to risk seeking with lotteries Ak. (See Figure 13). Player chooses lottery with higher risk and higher possible gain. For some size of disadvantage lottery An (lottery with highest gain) is chosen.
Figure 13  Case of big disadvantage and risk seeking behavior of player i
We conclude that prospect theory suggests that risk attitude changes depending on the initial advantage xi. Higher advantage corresponds to taking less risk (with minimum for A1), and lower advantage corresponds to taking more risk (with maximum for An). If we refer to prospect theory directly, then players wish to keep the gain and eliminate the loss. Gain/loss is a deviation from reference point, and such deviation is an advantage in our model. Thus such dependence between initial advantage and risk attitude is supported by the model and by prospect theory itself.
In some round t player reacts to initial advantage in this round (xt) by choosing lottery with potential gain at. Higher xt suggests that less risky lottery with less gain is chosen, therefore at should decrease with xt. We can test the existence of this behavior pattern derived from prospect theory.
The sign of relationship between xt and at is a crucial thing, and we use a linear regression model to get a sign in the relationship at(xt) = б + вxt . Model suggests that there is no such simple relationship as we wish to estimate. However, we are interested only in the sign of such relationship, so we can estimate a relationship at(xt).
Data contains 40 pairs of values xt and at, so t = 1..40 for each individual. We use ordinary least squares (OLS) method for estimating coefficient в in the relationship at(xt) for each player separately.
Let us calculate estimated values of slopes bi of relationship at(xt) for each player i using OLS formula:
Table 2 (See Attachments) reflects all estimated values of bi for i = 1..30. There are only three positive values of bi, while other 27 values are negative.
We use all 30 values of estimated slopes bi as one sample with 30 players. Such nonparametric test as one sample median test (Wilcoxon signedrank test) does not require assumptions about bi population distribution. And this test is used in order to test a null hypothesis H0: b = 0 against the alternative b ? 0.
The test yields Zvalue = 4.494, and pvalue > 0 which means that H0 is rejected at 1% significance level.
Average value of b is 0.165, and with such pvalue and rejected hypothesis, we can say that there is evidence for в to be negative. Thus, in each round t chosen level of risk measured by variable at negatively depends on the initial advantage measured by variable xt.
When risk attitude of player moves towards risk aversion (variable a decreases) while being ahead (variable x increases) and moves towards risk seeking (variable a increases) while falling back (variable x decreases).
Such dependence satisfies the prospect theory and its effects. Notice that these results are obtained even without knowing precise individual functions for players (e,g. value functions). Evidence suggest that players' behaviour patterns satisfy prospect theory. Thus, there are signals about realization and applicability of prospect theory in the competitive environment as described in the research model.
Conclusion
In the research, we come to following conclusions.
Theoretical component. Prospect theory is the most progressive and modern concept, which explains the phenomena of the processes taking place in decisionmaking under risk and uncertainty. We construct a model and suggest a way prospect theory can be applied to competitive environment in order to explain behaviour of individuals in such setting. However, solution of the model depends on the individual attributes of particular players.
Experimental component. Conducted experiment provides the evidence for existence of prospect theory effects in competitive environment. We confirm that risk attitude of players changes depending on their position with respect to rival's position (e.g. score). In a competition players become relatively more risk averse the more they are ahead of rivals and relatively more risk seeking the more they fall behind. Such experimental results are explained by prospect theory with an assumption that players consider rival's score as a reference point.
Application of the results. Our research provides a way risk attitude changes are explained in the competitive environment. Prospect theory can be applied to this setting, thus theoretical analysis with such powerful theory becomes available. An understanding of the decisionmaking process in a competition improves. Prospect theory can be then used for analysis and policymaking in such competitive contexts as education, politics, sports etc. For instance, prospect theory suggest that individual in the end of rating is more prone to the risk. Policymakers should be aware that such individual will participate in illegal activities such as cheating with higher probability than an individual on the top positions.
Further researches. The research can be a foundation for further process of studying prospect theory in competitive environment. In particular, model can be expanded for more players, or lag in perception of rival's score can be introduced. Both these cases can be more frequently seen in the real world. Application of prospect theory faces the problem of unknown individual attributes of particular players. Thus we suggest an introduction of special experiment stages that can help to determine necessary function. Different methods of data analysis such as conditional probit can be also applied in further analysis of prospect theory application in competitive environment under risk and uncertainty.
References
1. Bernstein J. The Investor's Quotient: The Psychology of Successful Investing in Commodities & Stocks. New York: Wiley, 1993.
2. Camerer C. Prospect Theory in the Wild: Evidence from the Field // Kahneman D., Tversky A. (cds.) Choices, Values, and Frames. P. 288300.
3. Edwards, W., Winterfeldt, D. On Cognitive Illusions and Their Implications. Southern California Law Review, 1986, Vol. 59, pp. 401451.
4. Henslin, J. Craps and Magic. American Journal of Sociology, 1967, Vol. 73, pp. 316330.
5. Kahneman D., Tversky A. Prospect theory: an analysis of decision under risk // Econometrica, 1979, v. 47, #2, pp. 263291.
6. Kahneman, D., Tversky, A. "Advances in prospect theory: Cumulative representation of uncertainty", 1992, Journal of Risk and Uncertainty 5 (4): 297323.
7. Kahneman, D., Slovic, P., & Tversky, A. “Judgment Under Uncertainty: Heuristics and Biases”, 1982, New York: Cambridge University Press.
8. Knight. Frank H. Risk, Uncertainty and Profit /Knight. Frank H. Washington, D.C.: Beard Books, 2002 . 447 p.
9. Levitt, Steven D. and John A. List, “Homo economicus evolves,” Science, February 15, 2008, 319(5865), pp. 909910.
10. Markov, A.A. Theory of algorithms. [Translated by Jacques J. SchorrKon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e. Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algorifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 6051085.]
11. McCormick, Iain A.; Frank H. Walkey, Dianne E. Green. "Comparative perceptions of driver ability A confirmation and expansion". Accident Analysis & Prevention (1986); 18(3) pp. 205208.
12. McFadden, D., “Conditional Logit Analysis of Qualitative Choice Behavior” in P. Zarembka (ed), Frontiers in Econometrics Academic Press, 1973, New York 105142.
13. Research on Judgment and Decision Making. / Ed. by W.M. Goldstein, R.M. Hogarth. Cambridge, 1997.
14. Rose McDermott, James H. Fowler, Oleg Smirnov. “On the Evolutionary Origin of Prospect Theory Preferences”, The Journal of Politics, Vol. 70, No. 2, April 2008, pp. 335350.
15. Shefrin H.M., Thaler R. An Economic Theory of Selfcontrol. NBER Working Paper 208, July 1978.
16. Strickland, L., Lewicki, R., Katz, A., Temporal Orientation and Perceived Control as Determanats of Risktaking. Journal of Experimental Social Psychology, 1966, Vol. 2, pp. 143151.
17. Strotz R. Myopia and Inconsistency in Dynamic Utility Maximization // Review of Economic Studies. Vol. 23. No.3. 19551956. pp. 165180.
18. Svenson, O. , “Are we all less risky and more skillful than our fellow drivers?"Acta Psychologica, 47 (2, February 1981): pp. 143148.
19. Wakker, Peter P., " Prospect Theory: For Risk and Ambiguity “, 2010, Cambridge: Cambridge University Press.
Appendix 1
Experiment form in English language
Your number: ____ 

Rival's number: ____ 

There are 4 different lotteries, which are available for you in each round of the game. Lotteries differ in sizes of gains and probabilities of success:
A: +1 point with probability 100% (6/6), it is not necessary to throw a dice;
B: +2 points with probability 50% (3/6), you need to roll “4” or more on a dice;
C: +3 points with probability 33% (2/6), you need to roll “5” or more on a dice;
D: +6 points with probability 17% (1/6), you need to roll “6” on a dice;
Your score 

Rival's score 

# 
A 
B 
C 
D 
# 
A 
B 
C 
D 

0 
 
 
 
 
0 
0 
30 

1 
31 

2 
32 

3 
33 

4 
34 

5 
35 

6 
36 

7 
37 

8 
38 

9 
39 

10 
40 

A 
B 
C 
D 
A 
B 
C 
D 

11 
41 

12 
42 

13 
43 

14 
44 

15 
45 

16 
46 

17 
47 

18 
48 

19 
49 

20 
50 

A 
B 
C 
D 
A 
B 
C 
D 

21 
51 

22 
52 

23 
53 

24 
54 

25 
55 

26 
56 

27 
57 

28 
58 

29 
59 

Appendix 2
Experiment form in Russian language
Ваш номер: ____ 

Номер оппонента: ____ 

В каждом раунде игры вам доступны 4 лотереи с разными размерами и вероятностями выигрыша:
A: +1 очко с вероятностью 100% (6/6), кубик можно не бросать;
B: +2 очка с вероятностью 50% (3/6), на кубике нужно выбросить “4” или более;
C: +3 очка с вероятностью 33% (2/6), на кубике нужно выбросить “5” или более;
D: +6 очков с вероятностью 17% (1/6), на кубике нужно выбросить “6”;
Ваш счёт 

Счёт оппонента 

# 
A 
B 
C 
D 
# 
A 
B 
C 
D 

0 
 
 
 
 
0 
0 
30 

1 
31 

2 
32 

3 
33 

4 
34 

5 
35 

6 
36 

7 
37 

8 
38 

9 
39 

10 
40 

A 
B 
C 
D 
A 
B 
C 
D 

11 
41 

12 
42 

13 
43 

14 
44 

15 
45 

16 
46 

17 
47 

18 
48 

19 
49 

20 
50 

A 
B 
C 
D 
A 
B 
C 
D 

21 
51 

22 
52 

23 
53 

24 
54 

25 
55 

26 
56 

27 
57 

28 
58 

29 
59 

Appendix 3
Photo of the process of conducting the experiment.
Appendix 4
Results of the experiment
Table 1  Results of the experiment for 30 players and 15 games played
game 
round 
player 
x 
a 
player 
x 
a 

game 
round 
player 
x 
a 
player 
x 
a 

1 
1 
1 
0 
1 
2 
0 
1 

2 
1 
3 
0 
1 
4 
0 
2 

1 
2 
1 
0 
1 
2 
0 
1 

2 
2 
3 
1 
1 
4 
1 
3 

1 
3 
1 
0 
2 
2 
0 
1 

2 
3 
3 
1 
2 
4 
1 
3 

1 
4 
1 
1 
2 
2 
1 
1 

2 
4 
3 
4 
3 
4 
4 
1 

1 
5 
1 
0 
1 
2 
0 
1 

2 
5 
3 
2 
3 
4 
2 
1 

1 
6 
1 
0 
1 
2 
0 
1 

2 
6 
3 
0 
3 
4 
0 
1 

1 
7 
1 
0 
1 
2 
0 
1 

2 
7 
3 
1 
3 
4 
1 
1 

1 
8 
1 
0 
2 
2 
0 
1 

2 
8 
3 
2 
3 
4 
2 
1 

1 
9 
1 
1 
2 
2 
1 
1 

2 
9 
3 
0 
3 
4 
0 
1 

1 
10 
1 
2 
3 
2 
2 
1 

2 
10 
3 
2 
3 
4 
2 
1 

1 
11 
1 
1 
3 
2 
1 
1 

2 
11 
3 
1 
3 
4 
1 
1 

1 
12 
1 
0 
3 
2 
0 
2 

2 
12 
3 
0 
1 
4 
0 
1 

1 
13 
1 
0 
1 
2 
0 
2 

2 
13 
3 
0 
1 
4 
0 
1 

1 
14 
1 
1 
1 
2 
1 
2 

2 
14 
3 
0 
1 
4 
0 
1 

1 
15 
1 
0 
2 
2 
0 
2 

2 
15 
3 
0 
1 
4 
0 
1 

1 
16 
1 
2 
1 
2 
2 
3 

2 
16 
3 
0 
1 
4 
0 
2 

1 
17 
1 
0 
1 
2 
0 
3 

2 
17 
3 
1 
2 
4 
1 
1 

1 
18 
1 
1 
1 
2 
1 
3 

2 
18 
3 
2 
2 
4 
2 
1 

1 
19 
1 
2 
6 
2 
2 
3 

2 
19 
3 
1 
2 
4 
1 
1 

1 
20 
1 
1 
1 
2 
1 
6 

2 
20 
3 
0 
2 
4 
0 
1 

1 
21 
1 
0 
1 
2 
0 
6 

2 
21 
3 
1 
3 
4 
1 
1 

1 
22 
1 
1 
1 
2 
1 
3 

2 
22 
3 
2 
3 
4 
2 
1 

1 
23 
1 
1 
1 
2 
1 
3 

2 
23 
3 
0 
3 
4 
0 
1 

1 
24 
1 
0 
1 
2 
0 
3 

2 
24 
3 
2 
6 
4 
2 
1 

1 
25 
1 
2 
1 
2 
2 
3 

2 
25 
3 
1 
6 
4 
1 
1 

1 
26 
1 
1 
1 
2 
1 
3 

2 
26 
3 
0 
6 
4 
0 
1 

1 
27 
1 
0 
1 
2 
0 
6 

2 
27 
3 
1 
1 
4 
1 
1 

1 
28 
1 
1 
1 
2 
1 
6 

2 
28 
3 
1 
1 
4 
1 
3 

1 
29 
1 
2 
1 
2 
2 
6 

2 
29 
3 
0 
1 
4 
0 
2 

1 
30 
1 
3 
1 
2 
3 
3 

2 
30 
3 
1 
1 
4 
1 
1 

1 
31 
1 
4 
1 
2 
4 
3 

2 
31 
3 
1 
1 
4 
1 
1 

1 
32 
1 
5 
1 
2 
5 
3 

2 
32 
3 
1 
1 
4 
1 
1 

1 
33 
1 
6 
1 
2 
6 
3 

2 
33 
3 
1 
1 
4 
1 
1 

1 
34 
1 
4 
2 
2 
4 
3 

2 
34 
3 
1 
2 
4 
1 
2 

1 
35 
1 
6 
2 
2 
6 
3 

2 
35 
3 
3 
1 
4 
3 
1 

1 
36 
1 
6 
2 
2 
6 
3 

2 
36 
3 
3 
1 
4 
3 
2 

1 
37 
1 
8 
1 
2 
8 
3 

2 
37 
3 
2 
2 
4 
2 
2 

1 
38 
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