# Учебная работа. The decision-making process of individuals in a competitive environment under risk and uncertainty

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

**Introduction**

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., 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.and objectives of the researchof 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.:theoretical aspects of the decision-making process of individuals under risk and uncertainty.prospect theory to the competitive environment with uncertainty elements and construct a relevant model.and conduct an experiment to test an application and effects of prospect theory to the competitive environment with uncertainty elements.the results, which reflect behavior of individuals in the decision-making process of individuals in a competitive environment under risk and uncertainty.

**Section 1. Literature review**

**1.1 Choice under 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.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: Σ (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.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.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. 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.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.are considered in terms of gains/losses, where gains/losses are deviations from some reference 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.a result, agents value functions have asymmetric S-shapes (See Figure 1).

1 — Prospect theory: value function (Kahneman, Daniel, and Amos Tversky (1979) Prospect Theory: An Analysis of Decision under Risk, Econometrica, XVLII, 263-291)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

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)

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. 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.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).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].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 real-world contexts are described with help of tournaments games. Researches include different types of games such as auctions.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.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.

**Section 2. Model**

**.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.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.round of the game consists of three stages.simultaneously make choices over a set of risky lotteries that may yield points.determine how many points they actually gain from lotteries chosen.update their scores.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.each lottery Ak we introduce notation for probability pk = a1/ak — probability of receiving ak points in lottery Ak.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. Players 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.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.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.refer to participating players as player 1 and player 2 together with following denotations for a chosen round (for example, round t):- score of player i at the beginning of the round, which is number of points accumulated by player i before current round.= 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.* — score of player i at the end of the round.* = 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.of player does not matter on its own, and higher advantage leads to higher probability of winning the game in the future.the whole game utility function of a player positively depends on the

**2.2 Application of prospect theory**

In particular, we are interested in the reference dependence effect. Individuals are concerned about deviations from some reference

3 — Application of prospect theory to the competition between two players

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).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.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.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.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 zero-sum game, and we right down such matrix for x1*.

Player 2A1A2p1 = 1p2 = a1/a21 — p2 = 1 — a1/a2+ a1+ a20Player 1A1p1 = 1+ a1x1x1+a1-a2x1+a1A2p2 = a1/a2+ a2x1+a2-a1x1x1+a21 — p2 = 1 — a1/a20x1-a1x1-a2x1Figure 4 — Matrix for advantage xi* for two lotteries

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 rivals score ∆y2. Player starts round in

Second, player can move to

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

5 — Values of function v1(x1*) for given value of ∆y2

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.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.

Player 2A1A2Player 1A1A2Figure 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:advantage xi;of lotteries Ak — variables a1 and a2;function vi and probability weighting function wi;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.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.#1of 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).

Player 2A1A2Player 1A1A2Figure 7 — Matrix for overall utility function u1 for two lotteries in the Example #1

us calculate approximate values of function ui.

Player 2A1A2Player 1A11.411.37A21.371.45Figure 8 — Matrix for approximate values of u1 in the Example #1

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.consider another example — Example #2.#23 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 4A1A2Player 3A11.411.37A21.371.30Figure 9 — Matrix for approximate values of u3 in the Example #2

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.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.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.

**Section 3. Experiment**

**.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.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.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.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. 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.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.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 rivals action until choices are made by both players. Then players determine how many points they actually gain and how scores change. Next round begins.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.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 rivals number (See Figure 10).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.

10 — First part of the form used for an experiment

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 the second column, when they get to round 30. Figure 11 illustrates beginning of the game log and first 6 rows.

11 — First 6 rows of the game log in the form used for an experiment

describe how players use game log with an example of round 1.the whole game players sit at the table against each other. When round 1 begins, each player does the following: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).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.until both players will make decisions, still covering 4 cells with a hand.what lottery is chosen.a dice.how many points lottery actually yield. Description of lotteries includes numbers players need to roll in order to gain points in a chosen lottery.the score. Score increases by a lottery gain, if roll is successful or lottery A is chosen.down the result in column 6.rival about the updated score.down rivals score in column 7.

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.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 t-1). 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.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):- 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;- 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., 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.the experiment there were 30 participants in the age group 17-23 years, including 16 women and 14 men. Average age was 19.3 years. All participants were either undergraduates (26) or graduates (4).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.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.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.of all 15 games with 40 rounds each were then transferred to the electronic form for further analysis in a specialized software.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.hypothesis testing is performed using one sample median test (Wilcoxon signed-rank 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**

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 decision-making process is independent in each round of the game. Also it is assumed that player bases a decision on players advantage in the beginning of the round, and no information from other rounds is used., from all available data we only need data on initial advantage and players 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).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.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 individuals preference between two alternatives is not affected by the introduction of a third alternative., 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.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.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.

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.

13 — Case of big disadvantage and risk seeking behavior of player i

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 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 β 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.use all 30 values of estimated slopes bi as one sample with 30 players. Such non-parametric test as one sample median test (Wilcoxon signed-rank 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.test yields Z-value = -4.494, and p-value → 0 which means that H0 is rejected at 1% significance level.value of b is -0.165, and with such p-value and rejected hypothesis, we can say that there is evidence for β 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).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**

the research, we come to following conclusions.component. Prospect theory is the most progressive and modern concept, which explains the phenomena of the processes taking place in decision-making 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.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 rivals 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 rivals score as a reference

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**Appendix 1**

Experiment form in English language

Your number: ____Rivals 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:: +1 Your scoreRival’s score

#ABCD#ABCD0—-00301312323334345356367378389391040ABCDABCD1141124213431444154516461747184819492050ABCDABCD215122522353245425552656275728582959

**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 счётСчёт оппонента

#ABCD#ABCD0—-00301312323334345356367378389391040ABCDABCD1141124213431444154516461747184819492050ABCDABCD215122522353245425552656275728582959

**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

gameroundplayerxaplayerxa gameroundplayerxaplayerxa11101201 2130140212101201 223114-1313102201 233-12413141-12211 243-4344115101201 253-2342116101201 2630340117101201 273-1341118102201 283-23421191122-11 293034011101232-21 2103234-211111132-11 2113134-11112103202 212301401113101202 2133014011141112-12 214301401115102202 2153014011161212-23 216301402117101203 2173-124111181112-13 2183-224211191262-23 2193-124111201-11216 220302401121101206 2213-134111221112-13 2223-234211231-11213 223303401124101203 2243264-211251-21223 2253164-111261-11213 226306401127101206 2273-114111281112-16 2283-114131291212-26 2293014021301312-33 2303114-111311412-43 2313114-111321512-53 2323114-111331612-63 2333114-111341422-43 2343124-121351622-63 2353314-311361622-63 2363314-321371812-83 2373224-221381612-63 2383234-211391712-73 2393134-111401812-83 240306401gameroundplayerxaplayerxa gameroundplayerxaplayerxa31502602 41701801325226-22 4270180133502603 43702801345-31633 447-12811355-21623 45702801365-41646 467128-11375-31636 47702801385-81686 487-12812395-71676 497-128123105-61666 4107128-123115-1116113 4117118-123125-1016103 4127218-223135-91693 4137318-333145-81683 4147418-433155-72673 4157518-533165-52653 4167618-633175-62663 4177418-463185-61663 4187518-563195-51656 4197628-663205-1036103 4207228-263215-1036103 4217228-263225-1336131 4227218-263235-1166111 4237-338313245-1266121 4247-138113255-1366131 4257-238213265-1466141 4267-338313275-1536152 4277-468413285-1236122 4287128-123295-1266121 4297128-123305-1366131 4307128-123315-1466141 4317318-333325-1566151 4327418-433335-1666161 4337518-533345-1766171 4347318-363355-1866182 4357418-463365-1866182 4367518-563375-2066202 4377618-663385-2266222 4387718-763395-2466241 4397818-863405-2566251 4407918-96gameroundplayerxaplayerxa gameroundplayerxaplayerxa519021002 61110112015292210-23 62110112015394110-43 63110112015495110-53 64110112015596110-63 65110312015697110-73 6611-1312115798110-83 6711-2312215896110-66 6811-3312315997110-76 6911-43124151098110-86 61011-53125151199210-96 61111-631261512911210-116 61211-761271513913110-136 61311-831281514914110-146 61411-931291515915110-156 61511-10312101516916110-166 61611-11312112517917110-176 61711-831282518918110-186 61811-831281519913110-136 61911-661261520914110-146 62011-761271521915110-156 62111-261221522916110-166 62211-361231523917310-176 62311-461241524911210-116 62411-561251525911210-116 62511-661261526913110-136 62611-761271527914110-146 62711-861281528915110-156 62811-961291529916110-166 62911-10612101530917110-176 63011-561251531912110-126 63111-66126253297110-76 63211-66126253398110-86 63311-86128253493110-36 63411-86128153594110-43 63511-96129153692110-23 63611-1061210153793110-33 63711-1161211153894110-43 63811-1261212153995110-53 63911-1361213154096110-66 64011-14612141gameroundplayerxaplayerxa gameroundplayerxaplayerxa7113031402 81150216027213-221421 82150216027313-321431 83150216027413-421441 84150216027513-531451 8515-2216227613-631461 8615-2216227713-431441 8715-2216227813-531451 8815-4216417913-631461 8915-52165171013-431441 81015-43164171113-231421 81115-53165171213011401 81215-63166171313011401 81315-73167171413011401 81415-53165171513011401 81515-33163171613011403 81615-431641717131114-13 81715-231621718132214-23 8181502160271913-121411 8191502160272013021402 82015-221621721132114-22 82115-321631722131114-12 82215-42164172313061402 82315-52165172413061402 82415-42164172513-261421 82515-32163172613-361431 82615-22162172713-461441 82715-12161272813-561451 82815-33163172913021402 82915-13161173013-221422 830151316-1273113-421442 831152116-2273213-421441 832153116-3273313-521451 833152116-2273413-431441 834151116-1273513-531451 835152116-2273613-631461 836153116-3273713-731471 837154116-4373813-531451 838155116-5373913-631461 839153116-3374013-731471 840154116-43gameroundplayerxaplayerxa gameroundplayerxaplayerxa9117011801 101190120029217011801 102191120-129317011801 103190120029417011801 104191120-129517011801 105192120-229617011801 106193120-329717011801 107192120-229817021801 108193120-3399171218-11 109194120-4391017021801 1010195120-52911171118-11 1011194120-42912171118-11 1012193220-32913171118-11 1013195220-52914171118-11 1014195220-52915171118-12 1015195220-5291617011801 1016195220-5391717011801 1017197120-7391817021801 1018198120-8391917-121811 1019199120-9392017021801 1020197220-7392117-121811 1021197120-7692217-221821 1022198120-8692317-121811 1023199120-9692417-221821 10241910120-10692517-121811 10251911120-11692617021802 10261912120-12692717-221822 1027197120-7692817021802 1028198120-8692917021802 1029199120-9693017021801 10301910120-106931171218-12 10311911120-116932173118-32 10321912120-126933172118-22 10331913120-136934173118-33 10341914120-146935174118-43 10351915120-156936175118-53 10361916120-166937176118-63 10371911120-116938174118-43 1038196120-66939175118-53 1039197120-76940173118-33 1040198120-86gameroundplayerxaplayerxa gameroundplayerxaplayerxa11121032202 1212302240111221032202 122231224-1211321-222223 123231324-1211421-532251 124231224-1111521-532252 1252302240111621-422242 12623-12241111721-432241 1272302240111821-532252 12823-12241111921-762272 12923-222421111021-922291 121023-332431111121-10122103 121123-132411111221-912293 121223-232421111321-832282 121323-322431111421-522251 121423-422441111521-662262 121523-322431111621-812282 121623-222421111721-962292 121723-322431111821-412241 121823-222421111921-332232 121923-122411112021-332232 122023022401112121-512251 1221231224-12112221-532253 1222233224-33112321-232222 1223235224-53112421-162212 1224235224-53112521-132212 1225237224-73112621-122213 1226237224-731127211322-12 1227239224-961128211322-11 12282311224-116112921012201 12292313224-136113021012202 12302315224-156113121-132212 12312315224-156113221-332232 12322315224-156113321-532252 12332317224-176113421-522252 12342319224-196113521-532252 12352319224-196113621-432242 12362321224-216113721-362232 12372315224-156113821-332232 12382315224-156113921-332232 12392317224-176114021-332231 12402319224-196gameroundplayerxaplayerxa gameroundplayerxaplayerxa13125012603 14127012802132251126-13 142271128-1213325-112613 143272128-2213425-322633 144273128-3213525-622662 145274128-4213625-622662 146273128-3213725-422642 147272128-2213825-432642 148273128-3213925-432642 149274128-42131025-432642 1410275128-53131125-632662 1411273128-33131225-832681 1412274128-43131325-632661 1413272128-23131425-732671 1414273128-33131525-862681 1415271128-12131625-962691 141627012802131725-10626102 141727-112811131825-10626102 141827-112811131925-10626102 141927-112811132025-12626121 142027-122811132125-13626131 142127012801132225-832681 142227022801132325-932692 1423271128-11132425-832682 1424271128-11132525-732672 1425271128-12132625-632662 142627012802132725-332632 142727-122812132825-332631 142827-122812132925-132612 142927-322831133025-332632 143027-222821133125022603 143127-122811133225022603 143227012801133325-322633 143327012801133425-322633 143427012801133525-422642 143527032801133625-632663 143627-122811133725-632661 143727012802133825-432641 1438271128-12133925-232622 1439272128-221340251326-13 1440273128-32gameroundplayerxaplayerxa151290230021522902300215329-22302215429-22302215529023002156292230-221572906300215829-26302215929-463042151029-463042151129-663061151229-763071151329-2630211514293630-311515292630-22151629063002151729-263022151829-463041151929-563052152029-563052152129-563051152229063001152329-163011152429-263021152529-363031152629-463041152729-563051152829033001152929-133011153029-233021153129023001153229-123011153329023002153429-2230221535290230021536290230021537292230-221538292230-221539292130-221540291130-12

**Appendix 5**

Analysis of the experiment

Table 2 — Values of estimated slopes bi in a relationship at(xt) = α + βxt for each player i

playerb playerb1-0,0059 16-0,19082-0,0637 17-0,14753-0,0527 18-0,32384-0,0343 19-0,01865-0,2676 20-0,36416-0,1408 210,00237-0,2215 220,02558-0,5184 23-0,01059-0,0004 24-0,268310-0,2032 25-0,319911-0,2575 26-0,1328120,0135 27-0,138113-0,1852 28-0,260914-0,1550 29-0,450315-0,1830 30-0,0652