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 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.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.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
point and classify outcomes as gains or losses. On second stage (evaluation),
individual associate each decision with certain value of utility.function v(z)
depends on the outcome z, and for each decision there is a set possible values
of this function. Utility function u equals to weighted sum of values of
possible outcomes. Prospect theory and its improved version (cumulative
prospect theory) suggests the non-linear probability weighting [6].indicates
that decision-makers have such probability weighting function w, that they
overweight small probabilities and underweight large probabilities (See Figure
2).
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. 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.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 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., 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. 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.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.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*.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.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.
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 2
|
|
A1
|
A2
|
|
p1 = 1
|
p2 = a1/a2
|
1 - p2 = 1 - a1/a2
|
|
+ a1
|
+ a2
|
0
|
Player 1
|
A1
|
p1 = 1
|
+ a1
|
x1
|
x1+a1-a2
|
x1+a1
|
|
A2
|
p2 = a1/a2
|
+ a2
|
x1+a2-a1
|
x1
|
x1+a2
|
|
|
1 - p2 = 1 - a1/a2
|
0
|
x1-a1
|
x1-a2
|
x1
|
Figure 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 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., player can move to point Ak*
with probability pk, so ak points are gained:
Second, player can move to point A0*
with probability (1-pk), so no points are gained:
Figure 5 illustrates changes in the
advantage for two lotteries and some value of ∆y2. Depending on the
lottery chosen (A1 or A2) there can be three different outcomes (A0*, A1*,
A2*), while round starts in A0. Notice, that if player chooses riskless lottery
A1, then there is always movement from point A0 to point A1*.
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 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: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 2
|
|
A1
|
A2
|
Player 1
|
A1
|
|
|
|
A2
|
|
|
Figure 7
- Matrix for overall utility function u1 for two
lotteries in the Example #1
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
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 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
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 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.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 rival’s 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 rival’s
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 player’s 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 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).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
individual’s 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 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.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.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)
= α + βxt . 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 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 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.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 decision-making 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.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.
. Camerer
C. Prospect Theory in the Wild: Evidence from the Field // Kahneman D., Tversky
A. (cds.) Choices, Values, and Frames. P. 288-300.
. Edwards,
W., Winterfeldt, D. On Cognitive Illusions and Their Implications. Southern
California Law Review, 1986, Vol. 59, pp. 401-451.
. Henslin,
J. Craps and Magic. American Journal of Sociology, 1967, Vol. 73, pp. 316-330.
. Kahneman
D., Tversky A. Prospect theory: an analysis of decision under risk //
Econometrica, 1979, v. 47, #2, pp. 263-291.
. Kahneman,
D., Tversky, A. "Advances in prospect theory: Cumulative representation of
uncertainty", 1992, Journal of Risk and Uncertainty 5 (4): 297-323.
. Kahneman,
D., Slovic, P., & Tversky, A. “Judgment Under Uncertainty: Heuristics and
Biases”, 1982, New York: Cambridge University Press.
. Knight.
Frank H. Risk, Uncertainty and Profit /Knight. Frank H.- Washington, D.C.:
Beard Books, 2002 .- 447 p.
. Levitt,
Steven D. and John A. List, “Homo economicus evolves,” Science, February 15,
2008, 319(5865), pp. 909-910.
. Markov,
A.A. Theory of algorithms. [Translated by Jacques J. Schorr-Kon 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 60-51085.]
. 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. 205-208.
. McFadden,
D., “Conditional Logit Analysis of Qualitative Choice Behavior” in P. Zarembka
(ed), Frontiers in Econometrics Academic Press, 1973, New York 105-142.
. Research
on Judgment and Decision Making. / Ed. by W.M. Goldstein, R.M. Hogarth.
Cambridge, 1997.
. 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. 335-350.
. Shefrin
H.M., Thaler R. An Economic Theory of Self-control. NBER Working Paper 208,
July 1978.
. Strickland,
L., Lewicki, R., Katz, A., Temporal Orientation and Perceived Control as
Determanats of Risk-taking. Journal of Experimental Social Psychology, 1966,
Vol. 2, pp. 143-151.
. Strotz
R. Myopia and Inconsistency in Dynamic Utility Maximization // Review of
Economic Studies. Vol. 23. No.3. 1955-1956. pp. 165-180.
. Svenson,
O. , “Are we all less risky and more skillful than our fellow
drivers?"Acta Psychologica, 47 (2, February 1981): pp. 143-148.
. Wakker,
Peter P., " Prospect Theory: For Risk and Ambiguity “, 2010, Cambridge:
Cambridge University Press.
Appendix 1
Experiment
form in English language
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Your number: ____
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Rival’s number: ____
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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
point with probability 100% (6/6), it is not necessary to throw a dice;: +2
points with probability 50% (3/6), you need to roll “4” or more on a dice;: +3
points with probability 33% (2/6), you need to roll “5” or more on a dice;: +6
points with probability 17% (1/6), you need to roll “6” on a dice;
#
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Appendix 2
Experiment form in Russian language
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Ваш номер: ____
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Номер оппонента: ____
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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”;
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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
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player
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a
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player
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a
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game
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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
|
1
|
6
|
1
|
2
|
-6
|
3
|
|
2
|
38
|
3
|
2
|
3
|
4
|
-2
|
1
|
1
|
39
|
1
|
7
|
1
|
2
|
-7
|
3
|
|
2
|
39
|
3
|
1
|
3
|
4
|
-1
|
1
|
1
|
40
|
8
|
1
|
2
|
-8
|
3
|
|
2
|
40
|
3
|
0
|
6
|
4
|
0
|
1
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
3
|
1
|
5
|
0
|
2
|
6
|
0
|
2
|
|
4
|
1
|
7
|
0
|
1
|
8
|
0
|
1
|
|
3
|
2
|
5
|
2
|
2
|
6
|
-2
|
2
|
|
4
|
2
|
7
|
0
|
1
|
8
|
0
|
1
|
|
3
|
3
|
5
|
0
|
2
|
6
|
0
|
3
|
|
4
|
3
|
7
|
0
|
2
|
8
|
0
|
1
|
|
3
|
4
|
5
|
-3
|
1
|
6
|
3
|
3
|
|
4
|
4
|
7
|
-1
|
2
|
8
|
1
|
1
|
|
3
|
5
|
5
|
-2
|
1
|
6
|
2
|
3
|
|
4
|
5
|
7
|
0
|
2
|
8
|
0
|
1
|
|
3
|
6
|
5
|
-4
|
1
|
6
|
4
|
6
|
|
4
|
6
|
7
|
1
|
2
|
8
|
-1
|
1
|
|
3
|
7
|
5
|
-3
|
1
|
6
|
3
|
6
|
|
4
|
7
|
7
|
0
|
2
|
8
|
0
|
1
|
|
3
|
8
|
5
|
-8
|
1
|
6
|
8
|
6
|
|
4
|
8
|
7
|
-1
|
2
|
8
|
1
|
2
|
|
3
|
9
|
5
|
-7
|
1
|
6
|
7
|
6
|
|
4
|
9
|
7
|
-1
|
2
|
8
|
1
|
2
|
|
3
|
10
|
5
|
-6
|
1
|
6
|
6
|
6
|
|
4
|
10
|
7
|
1
|
2
|
8
|
-1
|
2
|
|
3
|
11
|
5
|
-11
|
1
|
6
|
11
|
3
|
|
4
|
11
|
7
|
1
|
1
|
8
|
-1
|
2
|
|
3
|
12
|
5
|
-10
|
1
|
6
|
10
|
3
|
|
4
|
12
|
7
|
2
|
1
|
8
|
-2
|
2
|
|
3
|
13
|
5
|
-9
|
1
|
6
|
9
|
3
|
|
4
|
13
|
7
|
3
|
1
|
8
|
-3
|
3
|
|
3
|
14
|
5
|
-8
|
1
|
6
|
8
|
3
|
|
4
|
14
|
7
|
4
|
1
|
8
|
-4
|
3
|
|
3
|
15
|
5
|
-7
|
2
|
6
|
7
|
3
|
|
4
|
15
|
7
|
5
|
1
|
8
|
-5
|
3
|
|
3
|
16
|
5
|
-5
|
2
|
6
|
5
|
3
|
|
4
|
16
|
7
|
6
|
1
|
8
|
-6
|
3
|
|
3
|
17
|
5
|
-6
|
2
|
6
|
6
|
3
|
|
4
|
17
|
7
|
4
|
1
|
8
|
-4
|
6
|
|
3
|
18
|
5
|
-6
|
1
|
6
|
6
|
3
|
|
4
|
18
|
7
|
5
|
1
|
8
|
-5
|
6
|
|
3
|
19
|
5
|
-5
|
1
|
6
|
5
|
6
|
|
4
|
19
|
7
|
6
|
8
|
-6
|
6
|
|
3
|
20
|
5
|
-10
|
3
|
6
|
10
|
3
|
|
4
|
20
|
7
|
2
|
2
|
8
|
-2
|
6
|
|
3
|
21
|
5
|
-10
|
3
|
6
|
10
|
3
|
|
4
|
21
|
7
|
2
|
2
|
8
|
-2
|
6
|
|
3
|
22
|
5
|
-13
|
3
|
6
|
13
|
1
|
|
4
|
22
|
7
|
2
|
1
|
8
|
-2
|
6
|
|
3
|
23
|
5
|
-11
|
6
|
6
|
11
|
1
|
|
4
|
23
|
7
|
-3
|
3
|
8
|
3
|
1
|
|
3
|
24
|
5
|
-12
|
6
|
6
|
12
|
1
|
|
4
|
24
|
7
|
-1
|
3
|
8
|
1
|
1
|
|
3
|
25
|
5
|
-13
|
6
|
6
|
13
|
1
|
|
4
|
25
|
7
|
-2
|
3
|
8
|
2
|
1
|
|
3
|
26
|
5
|
-14
|
6
|
6
|
14
|
1
|
|
4
|
26
|
7
|
-3
|
3
|
8
|
3
|
1
|
|
3
|
27
|
5
|
-15
|
3
|
6
|
15
|
2
|
|
4
|
27
|
7
|
-4
|
6
|
8
|
4
|
1
|
|
3
|
28
|
5
|
-12
|
3
|
6
|
12
|
2
|
|
4
|
28
|
7
|
1
|
2
|
8
|
-1
|
2
|
|
3
|
29
|
5
|
-12
|
6
|
6
|
12
|
1
|
|
4
|
29
|
7
|
1
|
2
|
8
|
-1
|
2
|
|
3
|
30
|
5
|
-13
|
6
|
6
|
13
|
1
|
|
4
|
30
|
7
|
1
|
2
|
8
|
-1
|
2
|
|
3
|
31
|
5
|
-14
|
6
|
6
|
14
|
1
|
|
4
|
31
|
7
|
3
|
1
|
8
|
-3
|
3
|
|
3
|
32
|
5
|
-15
|
6
|
6
|
15
|
1
|
|
4
|
32
|
7
|
4
|
1
|
8
|
-4
|
3
|
|
3
|
33
|
5
|
-16
|
6
|
6
|
16
|
1
|
|
4
|
33
|
7
|
5
|
1
|
8
|
-5
|
3
|
|
3
|
34
|
5
|
-17
|
6
|
6
|
17
|
1
|
|
4
|
34
|
7
|
3
|
1
|
8
|
-3
|
6
|
|
3
|
35
|
5
|
-18
|
6
|
6
|
18
|
2
|
|
4
|
35
|
7
|
4
|
1
|
8
|
-4
|
6
|
|
3
|
36
|
5
|
-18
|
6
|
6
|
18
|
2
|
|
4
|
36
|
7
|
5
|
1
|
8
|
-5
|
6
|
|
3
|
37
|
5
|
-20
|
6
|
6
|
20
|
2
|
|
4
|
37
|
7
|
6
|
1
|
8
|
-6
|
6
|
|
3
|
38
|
5
|
-22
|
6
|
6
|
22
|
2
|
|
4
|
38
|
7
|
7
|
1
|
8
|
-7
|
6
|
|
3
|
39
|
5
|
-24
|
6
|
6
|
24
|
1
|
|
4
|
39
|
7
|
8
|
1
|
8
|
-8
|
6
|
|
3
|
40
|
5
|
-25
|
6
|
6
|
25
|
|
4
|
40
|
7
|
9
|
1
|
8
|
-9
|
6
|
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
5
|
1
|
9
|
0
|
2
|
10
|
0
|
2
|
|
6
|
1
|
11
|
0
|
1
|
12
|
0
|
1
|
|
5
|
2
|
9
|
2
|
2
|
10
|
-2
|
3
|
|
6
|
2
|
11
|
0
|
1
|
12
|
0
|
1
|
|
5
|
3
|
9
|
4
|
1
|
10
|
-4
|
3
|
|
6
|
3
|
11
|
0
|
1
|
12
|
0
|
1
|
|
5
|
4
|
9
|
5
|
1
|
10
|
-5
|
3
|
|
6
|
4
|
11
|
0
|
1
|
12
|
0
|
1
|
|
5
|
5
|
9
|
6
|
1
|
10
|
-6
|
3
|
|
6
|
5
|
11
|
0
|
3
|
12
|
0
|
1
|
|
5
|
6
|
9
|
7
|
1
|
10
|
-7
|
3
|
|
6
|
6
|
11
|
-1
|
3
|
12
|
1
|
1
|
|
5
|
7
|
9
|
8
|
1
|
10
|
-8
|
3
|
|
6
|
7
|
11
|
-2
|
3
|
12
|
2
|
1
|
|
5
|
8
|
9
|
6
|
1
|
10
|
-6
|
6
|
|
6
|
8
|
11
|
-3
|
3
|
12
|
3
|
1
|
|
5
|
9
|
9
|
7
|
1
|
10
|
-7
|
6
|
|
6
|
9
|
11
|
-4
|
3
|
12
|
4
|
1
|
|
5
|
10
|
9
|
8
|
1
|
10
|
-8
|
6
|
|
6
|
10
|
11
|
-5
|
3
|
12
|
5
|
1
|
|
5
|
11
|
9
|
9
|
2
|
10
|
-9
|
6
|
|
6
|
11
|
11
|
-6
|
3
|
12
|
6
|
1
|
|
5
|
12
|
9
|
11
|
2
|
10
|
-11
|
6
|
|
6
|
12
|
11
|
-7
|
6
|
12
|
7
|
1
|
|
5
|
13
|
9
|
13
|
1
|
10
|
-13
|
6
|
|
6
|
13
|
11
|
-8
|
3
|
12
|
8
|
1
|
|
5
|
14
|
9
|
14
|
1
|
10
|
-14
|
6
|
|
6
|
14
|
11
|
-9
|
3
|
12
|
9
|
1
|
|
5
|
15
|
9
|
15
|
1
|
10
|
-15
|
6
|
|
6
|
15
|
11
|
-10
|
3
|
12
|
10
|
1
|
|
5
|
16
|
9
|
16
|
1
|
10
|
-16
|
6
|
|
6
|
16
|
11
|
-11
|
3
|
12
|
11
|
2
|
|
5
|
17
|
9
|
17
|
1
|
10
|
-17
|
6
|
|
6
|
17
|
11
|
-8
|
3
|
12
|
8
|
2
|
|
5
|
18
|
9
|
18
|
1
|
10
|
-18
|
6
|
|
6
|
18
|
11
|
-8
|
3
|
12
|
8
|
1
|
|
5
|
19
|
9
|
13
|
1
|
10
|
-13
|
6
|
|
6
|
19
|
11
|
-6
|
6
|
12
|
6
|
1
|
|
5
|
9
|
14
|
1
|
10
|
-14
|
6
|
|
6
|
20
|
11
|
-7
|
6
|
12
|
7
|
1
|
|
5
|
21
|
9
|
15
|
1
|
10
|
-15
|
6
|
|
6
|
21
|
11
|
-2
|
6
|
12
|
2
|
1
|
|
5
|
22
|
9
|
16
|
1
|
10
|
-16
|
6
|
|
6
|
22
|
11
|
-3
|
6
|
12
|
3
|
1
|
|
5
|
23
|
9
|
17
|
3
|
10
|
-17
|
6
|
|
6
|
23
|
11
|
-4
|
6
|
12
|
4
|
1
|
|
5
|
24
|
9
|
11
|
2
|
10
|
-11
|
6
|
|
6
|
24
|
11
|
-5
|
6
|
12
|
5
|
1
|
|
5
|
25
|
9
|
11
|
2
|
10
|
-11
|
6
|
|
6
|
25
|
11
|
-6
|
6
|
12
|
6
|
1
|
|
5
|
26
|
9
|
13
|
1
|
10
|
-13
|
6
|
|
6
|
26
|
11
|
-7
|
6
|
12
|
7
|
1
|
|
5
|
27
|
9
|
14
|
1
|
10
|
-14
|
6
|
|
6
|
27
|
11
|
-8
|
6
|
12
|
8
|
1
|
|
5
|
28
|
9
|
15
|
1
|
10
|
-15
|
6
|
|
6
|
28
|
11
|
-9
|
6
|
12
|
9
|
1
|
|
5
|
29
|
9
|
16
|
1
|
10
|
-16
|
6
|
|
6
|
29
|
11
|
-10
|
6
|
12
|
10
|
1
|
|
5
|
30
|
9
|
17
|
1
|
10
|
-17
|
6
|
|
6
|
30
|
11
|
-5
|
6
|
12
|
5
|
1
|
|
5
|
31
|
9
|
12
|
1
|
10
|
-12
|
6
|
|
6
|
31
|
11
|
-6
|
6
|
12
|
6
|
2
|
|
5
|
32
|
9
|
7
|
1
|
10
|
-7
|
6
|
|
6
|
32
|
11
|
-6
|
6
|
12
|
6
|
2
|
|
5
|
33
|
9
|
8
|
1
|
10
|
-8
|
6
|
|
6
|
33
|
11
|
-8
|
6
|
12
|
8
|
2
|
|
5
|
34
|
9
|
3
|
1
|
10
|
-3
|
6
|
|
6
|
34
|
11
|
-8
|
6
|
12
|
8
|
1
|
|
5
|
35
|
9
|
4
|
1
|
10
|
-4
|
3
|
|
6
|
35
|
11
|
-9
|
6
|
12
|
9
|
1
|
|
5
|
36
|
9
|
2
|
1
|
10
|
-2
|
3
|
|
6
|
36
|
11
|
-10
|
6
|
12
|
10
|
1
|
|
5
|
37
|
9
|
3
|
1
|
10
|
-3
|
3
|
|
6
|
37
|
11
|
-11
|
6
|
12
|
11
|
1
|
|
5
|
38
|
9
|
4
|
1
|
10
|
-4
|
3
|
|
6
|
38
|
11
|
-12
|
6
|
12
|
12
|
1
|
|
5
|
39
|
9
|
5
|
1
|
10
|
-5
|
3
|
|
6
|
39
|
11
|
-13
|
6
|
12
|
13
|
1
|
|
5
|
40
|
9
|
6
|
1
|
10
|
-6
|
6
|
|
6
|
40
|
11
|
6
|
12
|
14
|
1
|
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
7
|
1
|
13
|
0
|
3
|
14
|
0
|
2
|
|
8
|
1
|
15
|
0
|
2
|
16
|
0
|
2
|
|
7
|
2
|
13
|
-2
|
2
|
14
|
2
|
1
|
|
8
|
2
|
15
|
0
|
2
|
16
|
0
|
2
|
|
7
|
3
|
13
|
-3
|
2
|
14
|
3
|
1
|
|
8
|
3
|
15
|
0
|
2
|
16
|
0
|
2
|
|
7
|
4
|
13
|
-4
|
2
|
14
|
4
|
1
|
|
8
|
4
|
15
|
0
|
2
|
16
|
0
|
2
|
|
7
|
5
|
13
|
-5
|
3
|
14
|
5
|
1
|
|
8
|
5
|
15
|
-2
|
2
|
16
|
2
|
2
|
|
7
|
6
|
13
|
-6
|
3
|
14
|
6
|
1
|
|
8
|
6
|
15
|
-2
|
2
|
16
|
2
|
2
|
|
7
|
7
|
13
|
-4
|
3
|
14
|
4
|
1
|
|
8
|
7
|
15
|
-2
|
2
|
16
|
2
|
2
|
|
7
|
8
|
13
|
-5
|
3
|
14
|
5
|
1
|
|
8
|
8
|
15
|
-4
|
2
|
16
|
4
|
1
|
|
7
|
9
|
13
|
-6
|
3
|
14
|
6
|
1
|
|
8
|
9
|
15
|
-5
|
2
|
16
|
5
|
1
|
|
7
|
10
|
13
|
-4
|
3
|
14
|
4
|
1
|
|
8
|
10
|
15
|
-4
|
3
|
16
|
4
|
1
|
|
7
|
11
|
13
|
-2
|
3
|
14
|
2
|
1
|
|
8
|
11
|
15
|
-5
|
3
|
16
|
5
|
1
|
|
7
|
12
|
13
|
0
|
1
|
14
|
0
|
1
|
|
8
|
12
|
15
|
-6
|
3
|
16
|
6
|
1
|
|
7
|
13
|
13
|
0
|
1
|
14
|
0
|
1
|
|
8
|
13
|
15
|
-7
|
3
|
16
|
7
|
1
|
|
7
|
14
|
13
|
0
|
1
|
14
|
0
|
1
|
|
8
|
14
|
15
|
-5
|
3
|
16
|
5
|
1
|
|
7
|
15
|
13
|
0
|
1
|
14
|
0
|
1
|
|
8
|
15
|
15
|
-3
|
3
|
16
|
3
|
1
|
|
7
|
16
|
13
|
0
|
1
|
14
|
0
|
3
|
|
8
|
16
|
15
|
-4
|
3
|
16
|
4
|
1
|
|
7
|
17
|
13
|
1
|
1
|
14
|
-1
|
3
|
|
8
|
17
|
15
|
-2
|
3
|
16
|
2
|
1
|
|
7
|
18
|
13
|
2
|
2
|
14
|
-2
|
3
|
|
8
|
18
|
15
|
0
|
2
|
16
|
0
|
2
|
|
7
|
19
|
13
|
-1
|
2
|
14
|
1
|
1
|
|
8
|
19
|
15
|
0
|
2
|
16
|
0
|
2
|
|
7
|
20
|
13
|
0
|
2
|
14
|
2
|
|
8
|
20
|
15
|
-2
|
2
|
16
|
2
|
1
|
|
7
|
21
|
13
|
2
|
1
|
14
|
-2
|
2
|
|
8
|
21
|
15
|
-3
|
2
|
16
|
3
|
1
|
|
7
|
22
|
13
|
1
|
1
|
14
|
-1
|
2
|
|
8
|
22
|
15
|
-4
|
2
|
16
|
4
|
1
|
|
7
|
23
|
13
|
0
|
6
|
14
|
0
|
2
|
|
8
|
23
|
15
|
-5
|
2
|
16
|
5
|
1
|
|
7
|
24
|
13
|
0
|
6
|
14
|
0
|
2
|
|
8
|
24
|
15
|
-4
|
2
|
16
|
4
|
1
|
|
7
|
25
|
13
|
-2
|
6
|
14
|
2
|
1
|
|
8
|
25
|
15
|
-3
|
2
|
16
|
3
|
1
|
|
7
|
26
|
13
|
-3
|
6
|
14
|
3
|
1
|
|
8
|
26
|
15
|
-2
|
2
|
16
|
2
|
1
|
|
7
|
27
|
13
|
-4
|
6
|
14
|
4
|
1
|
|
8
|
27
|
15
|
-1
|
2
|
16
|
1
|
2
|
|
7
|
28
|
13
|
-5
|
6
|
14
|
5
|
1
|
|
8
|
28
|
15
|
-3
|
3
|
16
|
3
|
1
|
|
7
|
29
|
13
|
0
|
2
|
14
|
0
|
2
|
|
8
|
29
|
15
|
-1
|
3
|
16
|
1
|
1
|
|
7
|
30
|
13
|
-2
|
2
|
14
|
2
|
2
|
|
8
|
30
|
15
|
1
|
3
|
16
|
-1
|
2
|
|
7
|
31
|
13
|
-4
|
2
|
14
|
4
|
2
|
|
8
|
31
|
15
|
2
|
1
|
16
|
-2
|
2
|
|
7
|
32
|
13
|
-4
|
2
|
14
|
4
|
1
|
|
8
|
32
|
15
|
3
|
1
|
16
|
-3
|
2
|
|
7
|
33
|
13
|
-5
|
2
|
14
|
5
|
1
|
|
8
|
33
|
15
|
2
|
1
|
16
|
-2
|
2
|
|
7
|
34
|
13
|
-4
|
3
|
14
|
4
|
1
|
|
8
|
34
|
15
|
1
|
1
|
16
|
-1
|
2
|
|
7
|
35
|
13
|
-5
|
3
|
14
|
5
|
1
|
|
8
|
35
|
15
|
2
|
1
|
16
|
-2
|
2
|
|
7
|
36
|
13
|
-6
|
3
|
14
|
6
|
1
|
|
8
|
36
|
15
|
3
|
1
|
16
|
-3
|
2
|
|
7
|
37
|
13
|
-7
|
3
|
14
|
7
|
1
|
|
8
|
37
|
15
|
4
|
1
|
16
|
-4
|
3
|
|
7
|
38
|
13
|
-5
|
3
|
14
|
5
|
1
|
|
8
|
38
|
15
|
5
|
1
|
16
|
-5
|
3
|
|
7
|
39
|
13
|
-6
|
3
|
14
|
6
|
1
|
|
8
|
39
|
15
|
3
|
1
|
16
|
-3
|
3
|
|
7
|
40
|
13
|
-7
|
3
|
14
|
7
|
1
|
|
8
|
40
|
15
|
4
|
1
|
16
|
-4
|
3
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
9
|
1
|
17
|
0
|
1
|
18
|
0
|
1
|
|
10
|
1
|
19
|
0
|
1
|
20
|
0
|
2
|
|
9
|
2
|
17
|
0
|
1
|
18
|
0
|
1
|
|
10
|
2
|
19
|
1
|
1
|
20
|
-1
|
2
|
|
9
|
3
|
17
|
0
|
1
|
18
|
0
|
1
|
|
10
|
3
|
19
|
0
|
1
|
20
|
0
|
2
|
|
9
|
4
|
17
|
0
|
1
|
18
|
0
|
1
|
|
10
|
4
|
19
|
1
|
1
|
20
|
-1
|
2
|
|
9
|
5
|
17
|
0
|
1
|
18
|
0
|
1
|
|
10
|
5
|
19
|
2
|
1
|
20
|
-2
|
2
|
|
9
|
6
|
17
|
0
|
1
|
18
|
0
|
1
|
|
10
|
6
|
19
|
3
|
1
|
20
|
-3
|
2
|
|
9
|
7
|
17
|
0
|
1
|
18
|
0
|
1
|
|
10
|
7
|
19
|
2
|
1
|
20
|
-2
|
2
|
|
9
|
8
|
17
|
0
|
2
|
18
|
0
|
1
|
|
10
|
8
|
19
|
3
|
1
|
20
|
-3
|
3
|
|
9
|
9
|
17
|
1
|
2
|
18
|
-1
|
1
|
|
10
|
9
|
19
|
4
|
1
|
20
|
-4
|
3
|
|
9
|
10
|
17
|
0
|
2
|
18
|
0
|
1
|
|
10
|
10
|
19
|
5
|
1
|
20
|
-5
|
2
|
|
9
|
11
|
17
|
1
|
1
|
18
|
-1
|
1
|
|
10
|
11
|
19
|
4
|
1
|
20
|
-4
|
2
|
|
9
|
12
|
17
|
1
|
1
|
18
|
-1
|
1
|
|
10
|
12
|
19
|
3
|
2
|
20
|
-3
|
2
|
|
9
|
13
|
17
|
1
|
1
|
18
|
-1
|
1
|
|
10
|
13
|
19
|
5
|
2
|
20
|
-5
|
2
|
|
9
|
14
|
17
|
1
|
1
|
18
|
-1
|
1
|
|
10
|
14
|
19
|
5
|
2
|
20
|
-5
|
2
|
|
9
|
15
|
17
|
1
|
1
|
18
|
-1
|
2
|
|
10
|
15
|
19
|
5
|
2
|
20
|
-5
|
2
|
|
9
|
16
|
17
|
0
|
1
|
18
|
0
|
1
|
|
10
|
16
|
19
|
5
|
2
|
20
|
-5
|
3
|
|
9
|
17
|
17
|
0
|
1
|
18
|
0
|
1
|
|
10
|
17
|
19
|
7
|
1
|
20
|
-7
|
3
|
|
9
|
18
|
17
|
0
|
2
|
18
|
0
|
1
|
|
10
|
18
|
19
|
8
|
1
|
20
|
-8
|
3
|
|
9
|
19
|
17
|
-1
|
2
|
18
|
1
|
1
|
|
10
|
19
|
19
|
9
|
1
|
20
|
-9
|
3
|
|
9
|
20
|
17
|
0
|
2
|
18
|
0
|
1
|
|
10
|
20
|
7
|
2
|
20
|
-7
|
3
|
|
9
|
21
|
17
|
-1
|
2
|
18
|
1
|
1
|
|
10
|
21
|
19
|
7
|
1
|
20
|
-7
|
6
|
|
9
|
22
|
17
|
-2
|
2
|
18
|
2
|
1
|
|
10
|
22
|
19
|
8
|
1
|
20
|
-8
|
6
|
|
9
|
23
|
17
|
-1
|
2
|
18
|
1
|
1
|
|
10
|
23
|
19
|
9
|
1
|
20
|
-9
|
6
|
|
9
|
24
|
17
|
-2
|
2
|
18
|
2
|
1
|
|
10
|
24
|
19
|
10
|
1
|
20
|
-10
|
6
|
|
9
|
25
|
17
|
-1
|
2
|
18
|
1
|
1
|
|
10
|
25
|
19
|
11
|
1
|
20
|
-11
|
6
|
|
9
|
26
|
17
|
0
|
2
|
18
|
0
|
2
|
|
10
|
26
|
19
|
12
|
1
|
20
|
-12
|
6
|
|
9
|
27
|
17
|
-2
|
2
|
18
|
2
|
2
|
|
10
|
27
|
19
|
7
|
1
|
20
|
-7
|
6
|
|
9
|
28
|
17
|
0
|
2
|
18
|
0
|
2
|
|
10
|
28
|
19
|
8
|
1
|
20
|
-8
|
6
|
|
9
|
29
|
17
|
0
|
2
|
18
|
0
|
2
|
|
10
|
29
|
19
|
9
|
1
|
20
|
-9
|
6
|
|
9
|
30
|
17
|
0
|
2
|
18
|
0
|
1
|
|
10
|
30
|
19
|
10
|
1
|
20
|
-10
|
6
|
|
9
|
31
|
17
|
1
|
2
|
18
|
-1
|
2
|
|
10
|
31
|
19
|
11
|
1
|
20
|
-11
|
6
|
|
9
|
32
|
17
|
3
|
1
|
18
|
-3
|
2
|
|
10
|
32
|
19
|
12
|
1
|
20
|
-12
|
6
|
|
9
|
33
|
17
|
2
|
1
|
18
|
-2
|
2
|
|
10
|
33
|
19
|
13
|
1
|
20
|
-13
|
6
|
|
9
|
34
|
17
|
3
|
1
|
18
|
-3
|
3
|
|
10
|
34
|
19
|
14
|
1
|
20
|
-14
|
6
|
|
9
|
35
|
17
|
4
|
1
|
18
|
-4
|
3
|
|
10
|
35
|
19
|
15
|
1
|
20
|
-15
|
6
|
|
9
|
36
|
17
|
5
|
1
|
18
|
-5
|
3
|
|
10
|
36
|
19
|
16
|
1
|
20
|
-16
|
6
|
|
9
|
37
|
17
|
6
|
1
|
18
|
-6
|
3
|
|
10
|
37
|
19
|
11
|
1
|
20
|
-11
|
6
|
|
9
|
38
|
17
|
4
|
1
|
18
|
-4
|
3
|
|
10
|
38
|
19
|
6
|
1
|
20
|
-6
|
6
|
|
9
|
39
|
17
|
5
|
1
|
18
|
-5
|
3
|
|
10
|
39
|
19
|
7
|
1
|
20
|
-7
|
6
|
|
9
|
40
|
17
|
3
|
1
|
18
|
-3
|
3
|
|
10
|
40
|
19
|
8
|
1
|
20
|
-8
|
6
|
|
game
|
round
|
player
|
x
|
a
|
x
|
a
|
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
11
|
1
|
21
|
0
|
3
|
22
|
0
|
2
|
|
12
|
1
|
23
|
0
|
2
|
24
|
0
|
1
|
|
11
|
2
|
21
|
0
|
3
|
22
|
0
|
2
|
|
12
|
2
|
23
|
1
|
2
|
24
|
-1
|
2
|
|
11
|
3
|
21
|
-2
|
2
|
22
|
2
|
3
|
|
12
|
3
|
23
|
1
|
3
|
24
|
-1
|
2
|
|
11
|
4
|
21
|
-5
|
3
|
22
|
5
|
1
|
|
12
|
4
|
23
|
1
|
2
|
24
|
-1
|
1
|
|
11
|
5
|
21
|
-5
|
3
|
22
|
5
|
2
|
|
12
|
5
|
23
|
0
|
2
|
24
|
0
|
1
|
|
11
|
6
|
21
|
-4
|
2
|
22
|
4
|
2
|
|
12
|
6
|
23
|
-1
|
2
|
24
|
1
|
1
|
|
11
|
7
|
21
|
-4
|
3
|
22
|
4
|
1
|
|
12
|
7
|
23
|
0
|
2
|
24
|
0
|
1
|
|
11
|
8
|
21
|
-5
|
3
|
22
|
5
|
2
|
|
12
|
8
|
23
|
-1
|
2
|
24
|
1
|
1
|
|
11
|
9
|
21
|
-7
|
6
|
22
|
7
|
2
|
|
12
|
9
|
23
|
-2
|
2
|
24
|
2
|
1
|
|
11
|
10
|
21
|
-9
|
2
|
22
|
9
|
1
|
|
12
|
10
|
23
|
-3
|
3
|
24
|
3
|
1
|
|
11
|
11
|
21
|
-10
|
1
|
22
|
10
|
3
|
|
12
|
11
|
23
|
-1
|
3
|
24
|
1
|
1
|
|
11
|
12
|
21
|
-9
|
1
|
22
|
9
|
3
|
|
12
|
12
|
23
|
-2
|
3
|
24
|
2
|
1
|
|
11
|
13
|
21
|
-8
|
3
|
22
|
8
|
2
|
|
12
|
13
|
23
|
-3
|
2
|
24
|
3
|
1
|
|
11
|
14
|
21
|
-5
|
2
|
22
|
5
|
1
|
|
12
|
14
|
23
|
-4
|
2
|
24
|
4
|
1
|
|
11
|
15
|
21
|
-6
|
6
|
22
|
6
|
2
|
|
12
|
15
|
23
|
-3
|
2
|
24
|
3
|
1
|
|
11
|
16
|
21
|
-8
|
1
|
22
|
8
|
2
|
|
12
|
16
|
23
|
-2
|
2
|
24
|
2
|
1
|
|
11
|
17
|
21
|
-9
|
6
|
22
|
9
|
2
|
|
12
|
17
|
23
|
-3
|
2
|
24
|
3
|
1
|
|
11
|
18
|
21
|
-4
|
1
|
22
|
4
|
1
|
|
12
|
18
|
23
|
-2
|
2
|
24
|
2
|
1
|
|
11
|
19
|
21
|
-3
|
3
|
22
|
3
|
2
|
|
12
|
19
|
23
|
-1
|
2
|
24
|
1
|
1
|
|
11
|
20
|
21
|
-3
|
3
|
22
|
3
|
2
|
|
12
|
20
|
23
|
0
|
2
|
24
|
0
|
|
11
|
21
|
21
|
-5
|
1
|
22
|
5
|
1
|
|
12
|
21
|
23
|
1
|
2
|
24
|
-1
|
2
|
|
11
|
22
|
21
|
-5
|
3
|
22
|
5
|
3
|
|
12
|
22
|
23
|
3
|
2
|
24
|
-3
|
3
|
|
11
|
23
|
21
|
-2
|
3
|
22
|
2
|
2
|
|
12
|
23
|
23
|
5
|
2
|
24
|
-5
|
3
|
|
11
|
24
|
21
|
-1
|
6
|
22
|
1
|
2
|
|
12
|
24
|
23
|
5
|
2
|
24
|
-5
|
3
|
|
11
|
25
|
21
|
-1
|
3
|
22
|
1
|
2
|
|
12
|
25
|
23
|
7
|
2
|
24
|
-7
|
3
|
|
11
|
26
|
21
|
-1
|
2
|
22
|
1
|
3
|
|
12
|
26
|
23
|
7
|
2
|
24
|
-7
|
3
|
|
11
|
27
|
21
|
1
|
3
|
22
|
-1
|
2
|
|
12
|
27
|
23
|
9
|
2
|
24
|
-9
|
6
|
|
11
|
28
|
21
|
1
|
3
|
22
|
-1
|
1
|
|
12
|
28
|
23
|
11
|
2
|
24
|
-11
|
6
|
|
11
|
29
|
21
|
0
|
1
|
22
|
0
|
1
|
|
12
|
29
|
23
|
13
|
2
|
24
|
-13
|
6
|
|
11
|
30
|
21
|
0
|
1
|
22
|
0
|
2
|
|
12
|
30
|
23
|
15
|
2
|
24
|
-15
|
6
|
|
11
|
31
|
21
|
-1
|
3
|
22
|
1
|
2
|
|
12
|
31
|
23
|
15
|
2
|
24
|
-15
|
6
|
|
11
|
32
|
21
|
-3
|
3
|
22
|
3
|
2
|
|
12
|
32
|
23
|
15
|
2
|
24
|
-15
|
6
|
|
11
|
33
|
21
|
-5
|
3
|
22
|
5
|
2
|
|
12
|
33
|
23
|
17
|
2
|
24
|
-17
|
6
|
|
11
|
34
|
21
|
-5
|
2
|
22
|
5
|
2
|
|
12
|
34
|
23
|
19
|
2
|
24
|
-19
|
6
|
|
11
|
35
|
21
|
-5
|
3
|
22
|
5
|
2
|
|
12
|
35
|
23
|
19
|
2
|
24
|
-19
|
6
|
|
11
|
36
|
21
|
-4
|
3
|
22
|
4
|
2
|
|
12
|
36
|
23
|
21
|
2
|
24
|
-21
|
6
|
|
11
|
37
|
21
|
-3
|
6
|
22
|
3
|
2
|
|
12
|
37
|
23
|
15
|
2
|
24
|
-15
|
6
|
|
11
|
38
|
21
|
-3
|
3
|
22
|
3
|
2
|
|
12
|
38
|
23
|
15
|
2
|
24
|
-15
|
6
|
|
11
|
39
|
21
|
-3
|
3
|
22
|
3
|
2
|
|
12
|
39
|
23
|
17
|
2
|
24
|
-17
|
6
|
|
11
|
40
|
21
|
-3
|
3
|
22
|
3
|
1
|
|
12
|
40
|
23
|
19
|
2
|
24
|
-19
|
6
|
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
game
|
player
|
x
|
a
|
player
|
x
|
a
|
|
13
|
1
|
25
|
0
|
1
|
26
|
0
|
3
|
|
14
|
1
|
27
|
0
|
1
|
28
|
0
|
2
|
|
13
|
2
|
25
|
1
|
1
|
26
|
-1
|
3
|
|
14
|
2
|
27
|
1
|
1
|
28
|
-1
|
2
|
|
13
|
3
|
25
|
-1
|
1
|
26
|
1
|
3
|
|
14
|
3
|
27
|
2
|
1
|
28
|
-2
|
2
|
|
13
|
4
|
25
|
-3
|
2
|
26
|
3
|
3
|
|
14
|
4
|
27
|
3
|
1
|
28
|
-3
|
2
|
|
13
|
5
|
25
|
-6
|
2
|
26
|
6
|
2
|
|
14
|
5
|
27
|
4
|
1
|
28
|
-4
|
2
|
|
13
|
6
|
25
|
-6
|
2
|
26
|
6
|
2
|
|
14
|
6
|
27
|
3
|
1
|
28
|
-3
|
2
|
|
13
|
7
|
25
|
-4
|
2
|
26
|
4
|
2
|
|
14
|
7
|
27
|
2
|
1
|
28
|
-2
|
2
|
|
13
|
8
|
25
|
-4
|
3
|
26
|
4
|
2
|
|
14
|
8
|
27
|
3
|
1
|
28
|
-3
|
2
|
|
13
|
9
|
25
|
-4
|
3
|
26
|
4
|
2
|
|
14
|
9
|
27
|
4
|
1
|
28
|
-4
|
2
|
|
13
|
10
|
25
|
-4
|
3
|
26
|
4
|
2
|
|
14
|
10
|
27
|
5
|
1
|
28
|
-5
|
3
|
|
13
|
11
|
25
|
-6
|
3
|
26
|
6
|
2
|
|
14
|
11
|
27
|
3
|
1
|
28
|
-3
|
3
|
|
13
|
12
|
25
|
-8
|
3
|
26
|
8
|
1
|
|
14
|
12
|
27
|
4
|
1
|
28
|
-4
|
3
|
|
13
|
13
|
25
|
-6
|
3
|
26
|
6
|
1
|
|
14
|
13
|
27
|
2
|
1
|
28
|
-2
|
3
|
|
13
|
14
|
25
|
-7
|
3
|
26
|
7
|
1
|
|
14
|
14
|
27
|
3
|
1
|
28
|
-3
|
3
|
|
13
|
15
|
25
|
-8
|
6
|
26
|
8
|
1
|
|
14
|
15
|
27
|
1
|
1
|
28
|
-1
|
2
|
|
13
|
16
|
25
|
-9
|
6
|
26
|
9
|
1
|
|
14
|
16
|
27
|
0
|
1
|
28
|
0
|
2
|
|
13
|
17
|
25
|
-10
|
6
|
26
|
10
|
2
|
|
14
|
17
|
27
|
-1
|
1
|
28
|
1
|
1
|
|
13
|
18
|
25
|
-10
|
6
|
26
|
10
|
2
|
|
14
|
18
|
27
|
-1
|
1
|
28
|
1
|
1
|
|
13
|
19
|
25
|
-10
|
6
|
26
|
10
|
2
|
|
14
|
19
|
27
|
-1
|
1
|
28
|
1
|
1
|
|
13
|
20
|
25
|
-12
|
6
|
26
|
12
|
1
|
|
14
|
20
|
27
|
-1
|
2
|
28
|
1
|
1
|
|
13
|
21
|
25
|
-13
|
26
|
13
|
1
|
|
14
|
21
|
27
|
0
|
1
|
28
|
0
|
1
|
|
13
|
22
|
25
|
-8
|
3
|
26
|
8
|
1
|
|
14
|
22
|
27
|
0
|
2
|
28
|
0
|
1
|
|
13
|
23
|
25
|
-9
|
3
|
26
|
9
|
2
|
|
14
|
23
|
27
|
1
|
1
|
28
|
-1
|
1
|
|
13
|
24
|
25
|
-8
|
3
|
26
|
8
|
2
|
|
14
|
24
|
27
|
1
|
1
|
28
|
-1
|
1
|
|
13
|
25
|
25
|
-7
|
3
|
26
|
7
|
2
|
|
14
|
25
|
27
|
1
|
1
|
28
|
-1
|
2
|
|
13
|
26
|
25
|
-6
|
3
|
26
|
6
|
2
|
|
14
|
26
|
27
|
0
|
1
|
28
|
0
|
2
|
|
13
|
27
|
25
|
-3
|
3
|
26
|
3
|
2
|
|
14
|
27
|
27
|
-1
|
2
|
28
|
1
|
2
|
|
13
|
28
|
25
|
-3
|
3
|
26
|
3
|
1
|
|
14
|
28
|
27
|
-1
|
2
|
28
|
1
|
2
|
|
13
|
29
|
25
|
-1
|
3
|
26
|
1
|
2
|
|
14
|
29
|
27
|
-3
|
2
|
28
|
3
|
1
|
|
13
|
30
|
25
|
-3
|
3
|
26
|
3
|
2
|
|
14
|
30
|
27
|
-2
|
2
|
28
|
2
|
1
|
|
13
|
31
|
25
|
0
|
2
|
26
|
0
|
3
|
|
14
|
31
|
27
|
-1
|
2
|
28
|
1
|
1
|
|
13
|
32
|
25
|
0
|
2
|
26
|
0
|
3
|
|
14
|
32
|
27
|
0
|
1
|
28
|
0
|
1
|
|
13
|
33
|
25
|
-3
|
2
|
26
|
3
|
3
|
|
14
|
33
|
27
|
0
|
1
|
28
|
0
|
1
|
|
13
|
34
|
25
|
-3
|
2
|
26
|
3
|
3
|
|
14
|
34
|
27
|
0
|
1
|
28
|
0
|
1
|
|
13
|
35
|
25
|
-4
|
2
|
26
|
4
|
2
|
|
14
|
35
|
27
|
0
|
3
|
28
|
0
|
1
|
|
13
|
36
|
25
|
-6
|
3
|
26
|
6
|
3
|
|
14
|
36
|
27
|
-1
|
2
|
28
|
1
|
1
|
|
13
|
37
|
25
|
-6
|
3
|
26
|
6
|
1
|
|
14
|
37
|
27
|
0
|
1
|
28
|
0
|
2
|
|
13
|
38
|
25
|
-4
|
3
|
26
|
4
|
1
|
|
14
|
38
|
27
|
1
|
1
|
28
|
-1
|
2
|
|
13
|
39
|
25
|
-2
|
3
|
26
|
2
|
2
|
|
14
|
39
|
27
|
2
|
1
|
28
|
-2
|
2
|
|
13
|
40
|
25
|
1
|
3
|
26
|
-1
|
3
|
|
14
|
40
|
27
|
3
|
1
|
28
|
-3
|
2
|
|
game
|
round
|
player
|
x
|
a
|
player
|
x
|
a
|
|
15
|
1
|
29
|
0
|
2
|
30
|
0
|
|
15
|
2
|
29
|
0
|
2
|
30
|
0
|
2
|
|
15
|
3
|
29
|
-2
|
2
|
30
|
2
|
2
|
|
15
|
4
|
29
|
-2
|
2
|
30
|
2
|
2
|
|
15
|
5
|
29
|
0
|
2
|
30
|
0
|
2
|
|
15
|
6
|
29
|
2
|
2
|
30
|
-2
|
2
|
|
15
|
7
|
29
|
0
|
6
|
30
|
0
|
2
|
|
15
|
8
|
29
|
-2
|
6
|
30
|
2
|
2
|
|
15
|
9
|
29
|
-4
|
6
|
30
|
4
|
2
|
|
15
|
10
|
29
|
-4
|
6
|
30
|
4
|
2
|
|
15
|
11
|
29
|
-6
|
6
|
30
|
6
|
1
|
|
15
|
12
|
29
|
-7
|
6
|
30
|
7
|
1
|
|
15
|
13
|
29
|
-2
|
6
|
30
|
2
|
1
|
|
15
|
14
|
29
|
3
|
6
|
30
|
-3
|
1
|
|
15
|
15
|
29
|
2
|
6
|
30
|
-2
|
2
|
|
15
|
16
|
29
|
0
|
6
|
30
|
0
|
2
|
|
15
|
17
|
29
|
-2
|
6
|
30
|
2
|
2
|
|
15
|
18
|
29
|
-4
|
6
|
30
|
4
|
1
|
|
15
|
19
|
29
|
-5
|
6
|
30
|
5
|
2
|
|
15
|
20
|
29
|
-5
|
6
|
30
|
5
|
2
|
|
15
|
21
|
29
|
-5
|
6
|
30
|
5
|
1
|
|
15
|
22
|
29
|
0
|
6
|
30
|
0
|
1
|
|
15
|
23
|
29
|
-1
|
6
|
30
|
1
|
1
|
|
15
|
24
|
29
|
-2
|
6
|
30
|
2
|
1
|
|
15
|
25
|
29
|
-3
|
6
|
30
|
3
|
1
|
|
15
|
26
|
29
|
-4
|
6
|
30
|
4
|
1
|
|
15
|
27
|
29
|
-5
|
6
|
30
|
5
|
1
|
|
15
|
28
|
29
|
0
|
3
|
30
|
0
|
1
|
|
15
|
29
|
29
|
-1
|
3
|
30
|
1
|
1
|
|
15
|
30
|
29
|
-2
|
3
|
30
|
2
|
1
|
|
15
|
31
|
29
|
0
|
2
|
30
|
0
|
1
|
|
15
|
32
|
29
|
-1
|
2
|
30
|
1
|
1
|
|
15
|
33
|
29
|
0
|
2
|
30
|
0
|
2
|
|
15
|
34
|
29
|
-2
|
2
|
30
|
2
|
2
|
|
15
|
35
|
29
|
0
|
2
|
30
|
0
|
2
|
|
15
|
36
|
29
|
0
|
2
|
30
|
0
|
2
|
|
15
|
37
|
29
|
2
|
2
|
30
|
-2
|
2
|
|
15
|
38
|
29
|
2
|
2
|
30
|
-2
|
2
|
|
15
|
39
|
29
|
2
|
1
|
30
|
-2
|
2
|
|
15
|
40
|
29
|
1
|
1
|
30
|
-1
|
2
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Appendix 5
Analysis of the experiment
Table 2
- Values of estimated slopes bi in a relationship
at(xt) = α + βxt for each player i
player
|
b
|
|
player
|
b
|
1
|
-0,0059
|
|
16
|
-0,1908
|
2
|
-0,0637
|
|
17
|
-0,1475
|
3
|
-0,0527
|
|
18
|
-0,3238
|
4
|
-0,0343
|
|
19
|
-0,0186
|
5
|
-0,2676
|
|
20
|
-0,3641
|
6
|
-0,1408
|
|
21
|
0,0023
|
-0,2215
|
|
22
|
0,0255
|
8
|
-0,5184
|
|
23
|
-0,0105
|
9
|
-0,0004
|
|
24
|
-0,2683
|
10
|
-0,2032
|
|
25
|
-0,3199
|
11
|
-0,2575
|
|
26
|
-0,1328
|
12
|
0,0135
|
|
27
|
-0,1381
|
13
|
-0,1852
|
|
28
|
-0,2609
|
14
|
-0,1550
|
|
29
|
-0,4503
|
15
|
-0,1830
|
|
30
|
-0,0652
|