Download A Typology of Players: between Instinctive and Contemplative

ATypologyofPlayers:betweenInstinctive
andContemplative
ArielRubinstein*
TelAvivUniversity
and
NewYorkUniversity
July2014
Abstract
A new typology of players is proposed, in which actions in a game are classified as instinctive or
contemplative. A player’s type is determined by the probability that he chooses a contemplative
action. In order to test this typology, a set of 11 games was analyzed in which the classification
of actions as instinctive or contemplative is relatively natural. The classification is shown to be
supported by response time data, where instinctive (contemplative) actions are correlated with
fast (slow) responses. A subject's contemplative index (CI) is defined as the proportion of games
in which he chose a contemplative action. It is found that for each of the 11 games, the CI in the
other ten games is a good predictor of whether he chooses a contemplative action in that game.
The CI is used to analyze the choices in several additional games.
Key words: Response Time, Game Theory, Typology, Instinctive, Contemplative.
* My thanks to Hadar Binsky for his help in analyzing the data, to Eli Zvuluny for twelve years
of dedicated assistance in building and maintaining the site http://gametheory.tau.ac.il and to
Ayala Arad for helpful comments.
I acknowledge financial support from ERC grant 269143.
1 | P a g e 1. Introduction
The goal of the paper is to suggest a new typology of decision makers in strategic situations. A
"type" is defined as a category of individuals with common characteristics. In both the theoretical
and experimental literature, these common characteristics constitute a mode of behavior often
described by a distinct preference relation or a procedure of choice. Thus, for example, an agent
is a type 1 in the k-level literature if he always maximizes his expected payoff as a best response
to what he perceives as level 0 behavior. The "crazy" type in the repeated chain store paradox
game is an individual who enjoys fighting and will do so in all circumstances. The impatient
type in bargaining is one who utilizes a low discount factor.
Economic models are generally characterized by a rigid definition of types, reflecting the view
that individuals are "programmed" to act according to clear-cut procedures or a well-defined
preference relation. This, however, appears to be counterfactural to reality. Consider, for
example, the classification of people as "good” or “evil”. A good person does not always choose
the good action and the evil person does not always choose the evil action. The distinction
between good and evil people is based on the classification of actions into good and evil. A good
person is one who chooses the course of action labeled as “good” significantly more often than
an evil person and vice versa.
The proposed typology is based on the classification of actions in a game as instinctive or
contemplative. A person's type is determined by the probability that he will choose a
contemplative action. Thus, a player will be classified as type p if the probability of him
choosing a contemplative action is p.
The main body of the paper consists of identifying the typology for a specific dataset. The first
part of the paper (Section 3) provides the groundwork for defining the typology based on eleven
games available on the abovementioned site. The strategies in each of the game are partitioned
into instinctive and contemplative categories. The classification of actions as instinctive or
contemplative will be based on a combination of intuition and response time. As in Rubinstein
(2007, 2013), it is suggested that response time is a simple, cheap and attractive indicator of
whether the strategy used in a game or a decision problem is the result of an instinctive or
cognitive process. This approach is in line with Kahneman (2011) who classified quick and
instinctive responses as being the result of system I processes and slower contemplative
responses as being the result of system II processes.
The second part of the paper (Section 4), examines the effectiveness of the proposed typology by
using the data for ten of the games to predict the choice in the eleventh. It will be demonstrated
that the contemplative index (CI) of a player (based on any of the ten-game sets) is a useful
predictor of the probability that the player will choose a contemplative action in the eleventh
game.
2 | P a g e The third part of the paper (Section 5) analyzes some additional games in light of the new
typology. The distinction between instinctive and contemplative actions in these examples is not
as clear but is nonetheless informative for interpreting behavior.
Before arriving at this typology, two alternative systems were considered. In the first, clustering
techniques were used to find correlations between the action taken by a player in one game and
the action taken by him in another game. Based on the results, one can surmise that it is rare to
find tight correlations between the behavior of a subject in two different games, unless the games
are very similar. It appears not unreasonable to conclude that a player’s strategies in two
different games (with the same preference on the consequences) are independent of one another.
The second alternative was more promising and was proposed and applied in Rubinstein (2013)
in a non-strategic context. This typology classifies people as (relatively) fast or slow. For each
subject and each problem, a local rank was defined as the fraction of subjects who answered the
question faster than he did. For each subject in the dataset who answered at least 9 problems (the
median number of problems answered by a subject was 15), a “Global Index” (GI) was defined
as the median of his local ranks. Thus, a high global rank would indicate a systematically slow
subject while a low global rank would indicate hastiness. Section 6 presents results regarding
the connection between GI and the choice of contemplative actions.
Comment on the Literature: The use of response time to open the “black box” of decision
making is well-established in psychology and goes back 150 years to the work of Franciscus
Cornelis Donders (see Donders (1969)). Psychologists, however, generally study response time
in contexts where it is measured in fractions of a second. In the context of choice problems and
game situations, response time measures the time spent reflecting on and thinking about a
decision and is measured in dozens of seconds and even minutes.
For reasons beyond my understanding, economists were hostile to the use of response time until
recently, when it suddenly became a legitimate and popular tool. Currently, response time is used
in the literature for a number of purposes (see Spiliopoulos and Ortmann (2014) for a recent
classification of the RT literature). In this study, response time is used as follows:
(i)
(ii)
(iii)
3 | P a g e To interpret the meaning of choice in games. Previous works that used RT in this
way include: Rubinstein (2007) (for a variety of games), Lotito, Migheli, Ortona
(2011) (a public goods game), Agranov, Caplin and Tergiman (2012) (guessing
games), Arad and Rubinstein (2012) (General Blotto Tournament), and Brañas-Garza,
Meloso and Miller (2012) and Hertwig, Fischbacher and Bruhin (2013) (the
ultimatum game).
To serve as a measure for “predicting” the behavior of players in a game (see Clithero
and Rangel (2013), Rubinstein (2013) and Schotter and Trevino (2013)).
As a basis for defining a typology of players. In the last part of the paper, response
time will also be used to define a typology of players as fast or slow. This attempt is
an application of Rubinstein (2008, 2013).
2. The Data
This type of research requires a large number of subjects. For the current study, this was
accomplished using the data accumulated on the didactic website gametheory.tau.ac.il. The pool
of subjects is diverse relative to the standard pools of subjects used in experimental game theory.
Subjects were students in game theory courses from around the world. As of the beginning of
2014, the site had 50,000 users from 40 countries (some as “marginal” as China and India…).
More than half were from the US, Switzerland, UK, Colombia, Argentina and the Slovak
Republic. Needless to say, the sample is not representative of the “world” in any serious way.
The site contains a bank of game-theoretic and decision-theoretic problems. Teachers, most of
whom teach game theory courses, register on the site, assemble sets of problems and assign them
to their students. Students respond to the problems anonymously. Teachers have access to
statistics summarizing the choices of their students and a comparison to statistics for all other
respondents. The website records both the students’ answers and their response times.
The most commonly voiced criticisms of research of this type are the non-laboratory setting, in
which there is no control over subjects, and the lack of monetary incentives. Rubinstein (2013)
responded to these criticisms and also noted that statistical tests were unnecessary in view of the
clear results obtained.
Each problem is a description of a hypothetical game. A subject responds to the problem by
specifying his anticipated behavior in the role of one of the players. For each problem, the
responses of the quickest 5% of the subjects were removed (since it is clear that they randomly
chose an answer). Each problem is given two identifiers: a name and the serial number in the
gametheory.tau.ac.il system (marked with #). (The serial numbers make it easier to use the
database.)
Response time (RT) is measured as the number of seconds from when a problem is sent to a
subject until his response is recorded by the server in Tel Aviv. Given the speed of
communication, we can treat this as the time the problem is on the subject’s screen. A
commonly used indicator of speed of response is Median Response Time (MRT).
Another commonly used graphical tool is the response time cdf. For each problem, the set of
alternatives is divided into categories (which are often singletons) and for each category ‫ܥ‬, ‫ܨ‬஼ ሺ‫ݐ‬ሻ
is the proportion of subjects that chose an alternative in ‫ ܥ‬and who responded within ‫ ݐ‬seconds.
Rubinstein (2007, 2013) found that the graphs of the response time cdfs display remarkable
regularities: (1) They have a common shape that resembles an inverse Gaussian or lognormal
distribution. (2) The response time cdfs (given a particular problem) are almost always ordered
by the “first-order stochastic domination” relation. For any two exclusive categories C and D, if
‫ܨ‬஼ ሺ‫ݐ‬ሻ ൒ ‫ ܨ‬஽ ሺ‫ݐ‬ሻ for all t, then we say that the C-choosers respond faster than the D-choosers.
4 | P a g e 3. Eleven games
This section contains the basic results for eleven games. Results for some of the games were
reported in Rubinstein (2007) though with much smaller samples. In each game, there is a
natural classification of the actions as either instinctive or contemplative (notated in red). In
some of the cases, the instinctive actions are associated with salience. Starting with Thomas
Schelling, researchers have been searching for a unified theory to explain the concept of salience
but we still lack such a general theory. As a result, the k-level reasoning literature has no theory
to explain the level-0 strategy and either relates to it as a uniform random strategy or justifies it
on intuitive grounds, as I do. I don't have a theory of my own to explain the classification, which
is currently based on my own intuition. However, the classification is not entirely subjective and
will be shown to be supported by evidence: contemplative actions are associated with
significantly higher RT than the instinctive ones.
3.1. Zero-Sum Games (#15, #145 (16+144))
The following simple zero-sum game was presented to students in the form of a bi-matrix where
payoffs were presented as numbers, without specifying their interpretation (students in game
theory courses are, of course, familiar with this kind of presentation).
1
Percent
62%
38%
MRT
41s
57s
0.9
0.8
0.7
Frequencies
n=4715
T
B
0.6
0.5
0.4
0.3
0.2
T
B
0.1
0
0
50
100
150
200
Response Time
250
300
Table 1: A Zero-Sum Game
#15: You are Player 1 in the following game:
Player 2
L
R
T
2,-2
0,0
Player 1
B
0,0
1,-1
Imagine that Player 2 is an anonymous player. What will you play?
The subjects (n=4715) “played” the game in the role of the row player. Whether one interprets
the payoffs as monetary payoffs or vNM utilities, Nash equilibrium predicts that more players
will play B than T. However, the action T was the more popular choice (62%). The action T is
the instinctive one due to the high payoff associated with it while B is the contemplative choice
5 | P a g e since it requires thinking about the other player’s move. Indeed, the RT of T (41s) is much lower
than that of B (57s).
Comment: Two other versions of the game appear on the site: In #16, subjects were asked to
play the game in the role of the column player (i.e. facing negative numbers). In #144, the
column and row roles were reversed and subjects were asked to play the game as the row player
(again facing negative payoffs). Herein the actions will be referred as "-2" and "-1" (students
were presented with actions denoted by letters).
#16: You are Player 2 in the following game:
Player 1
T
B
Player 2
L
R
2,-2
0,0
0,0
1,-1
#144: You are Player 1 in the following game:
Player 1
T
B
Player 2
L
R
-2,2
0,0
0,0
-1,1
There was almost no difference in the behavior of subjects in these two variants: 84-87% of the
subjects played "-1" (includes only subjects who did not play #15). Therefore, judging only from
the observed behavior one can conclude that the framing of the zero-sum game (whether the
player who faces negative payoffs is the row player or the column player) is immaterial.
However, the response time data raises the possibility that the choice of “-1” has a somewhat
different meaning in each of the two versions. When playing as a row player (#144), with n=653,
the action "-1" had the quickest response time (44s vs. 54s), probably reflecting the participants’
instinctive deterrence from the payoff “-2”. When playing as a column player (#16), with n=734,
the choice of “-1” had the slowest response time (47s. vs. 39s). This finding is consistent with the
hypothesis that column players start to analyze the game from the point of view of the row
player. When played as a row player, the choice of “-1” is more instinctive, reflecting the
deterrence from the potential loss of “-2”. When playing as the column player, the choice of “-1”
might also be his best response given the player’s prediction that the row player will choose the
instinctive choice of “2”, which is an expression of contemplative thinking.
6 | P a g e 3.2. Hoteling’s Main Street Game (#68)
The following is a three-player discrete variant of Hoteling (1929)’s Main Street Game with
seven locations ordered on a line. 1
0.9
0.8
0.7
Frequencies
n=8329 Percent MRT
1
4%
46s
2
8%
79s
3
16%
72s
4
43%
42s
5
14%
64s
6
8%
65s
7
6%
44s
0.6
0.5
1
2
3
4
5
6
7
0.4
0.3
0.2
0.1
0
0
50
100
150
200
Response Time
250
300
Table 2: Hoteling’s Main Street Game
Imagine you are the manager of a chain of cafes competing with two other similar chains. Each of you is
about to rent a shop in one of the 7 new identical huge apartment buildings standing along a beach strip.
Once each of you knows exactly where the other two competitors locate, it will be too late to move to
another location. You expect that the customers (the residents in the 7 buildings) will not distinguish
between the three cafes and will pick the one which is closest to their home. In which building (a number
between 1 and 7) will you locate your cafe?
The game’s unique symmetric Nash equilibrium strategy assigns probabilities (0.4, 0.2, 0.4) to
positions 3, 4 and 5, respectively. The results (for n=8329) show that 43% chose the middle
position, which is double the equilibrium prediction (in contrast, in the two-player version of the
game, which appears on the site (#71), 69% of the 10,336 subjects chose the middle position).
Only 10% of the subjects chose the dominated actions 1 and 7. The choice of 4 is intuitively the
instinctive action. In the analysis, the non-contemplative actions were combined with the
dominated actions 1 and 7. And indeed the choices 1, 4 and 7 are all associated with much lower
RT than the four actions 2,3,5,6, which are classified as contemplative.
3.3. The Two-Contest Game (#66)
This game was suggested in Huberman and Rubinstein (2000). Subjects were asked to choose
one of two contests. In each one, subjects compete by guessing, as closely as possible, the
outcome of either 20 tosses of a coin or 20 rolls of a die.
7 | P a g e 1
n=1901 Percent MRT
Coin
68%
71s
Die
32%
88s
0.9
0.8
Frequencies
0.7
0.6
0.5
0.4
0.3
0.2
Coin
Die
0.1
0
0
50
100
150
200
Response Time
250
300
Table 3: The Two-Contest Game
Imagine you are participating in a game with over 200 participants worldwide. Each participant
chooses to compete in one of two contests. In contest A, each contestant guesses the outcomes of 20
coin flips (heads or tails). In contest B, each contestant guesses the outcomes of 20 rolls of a die
(i.e., each of the twenty guesses is a number 1, 2, 3, 4, 5 or 6). Each contest will be conducted
independently. In each contest, you will be competing against people who, like you, chose that
contest. After the guesses of all the participants are collected, a computer will simulate a series of 20
coin flips for contest A and a series of 20 rolls of a die for contest B. The winner of each contest
will be the person with the most correct guesses. (In the case of a tie, the winner will be chosen by a
lottery among those with the most correct guesses.) I choose to participate in:
A rational player would choose the contest he believes will attract fewer subjects. The instinctive
choice is the “easiest” task (i.e. the coin toss). Out of 1901 subjects, 68% chose “coin toss” and
indeed their response time was much lower than for those who chose “roll of the die” (71s vs.
88s). Thus, the choice of “coin toss” is classified as instinctive and “roll of the die” as
contemplative.
3.4. Relying on other player’s rationality (#3)
1
Percent
60%
40%
MRT
56s
51s
0.9
0.8
0.7
Frequencies
n=13524
A
B
0.6
0.5
0.4
0.3
0.2
A
B
0.1
0
0
50
100
150
200
Response Time
250
300
Table 4: Relying on the rationality of another player
You are player 1 in a two-person game with the following monetary payoff matrix:
Player 2
A
B
A
5,5
-100,4
Player 1
B
0,1
0,0
What will you play?
8 | P a g e In this game, there is a common interest for the players to reach the outcome (A, A). However,
player 1 will suffer a large loss if player 2 does not play rationally. Thus, B is the instinctive
action in this game. The choice of A requires that the player trusts the other player to play
rationally and therefore is classified as contemplative. Among 13,524 subjects, 60% chose A.
The classification of A as contemplative is supported by its MRT (56s), which is somewhat
longer than that of the play-it-safe option B (51s).
3.5. Successive Elimination (#4)
1
Percent
4%
27%
39%
30%
MRT
77s
153s
85s
89s
0.9
0.8
0.7
Frequencies
n=13399
A
B
C
D
0.6
0.5
0.4
0.3
A
B
C
D
0.2
0.1
0
0
50
100
150
200
Response Time
250
300
Table 5: Successive Elimination
You are player 1 in a two-person game with the following payoff matrix:
Player 2
A
B
C
D
A
5,2
2,6
1,4
0,4
B
0,0
3,2
2,1
1,1
Player 1
C
7,0
2,2
1,5
5,1
D
9,5
1,3
0,2
4,8
What will you play?
This game is designed to determine whether subjects follow a process of successive elimination
of dominated strategies. Only 27% of the 13,399 subjects chose B, the survivor of the successive
elimination of weakly dominated strategies, and their MRT is extremely high (153s). Therefore,
the choice in B is certainly not instinctive. The actions C and D are instinctive since their
relatively high payoffs attract the attention of subjects. Indeed, the response times of C and D are
much lower (85-89s) than that of B which is classified as contemplative. The action A is
dominated by C and the RT of the 4% of subjects who chose A is the lowest among the four
choices.
9 | P a g e 3.6. The Dictator Game (#195 (187+188))
Subjects were randomly assigned to play one of two versions of the Dictator Game with $100
(virtual money) “on the table”. In #187, they were asked to decide how much to give to the other
person; in #188 they decided how much to keep for themselves.
#187
Percent
24%
12%
23%
36%
5%
MRT
32s
36s
34s
28s
28s
n=2286
100
99
51-98
50
0-49
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Frequencies
Frequencies
n=2345
0
1
2-49
50
51-100
0.5
0.4
0
1
2-49
50
51-100
0.3
0.2
0.1
0
0
50
100
150
200
Response Time
#188
Percent
24%
11%
21%
40%
3%
250
MRT
33s
30s
33s
29s
32s
0.5
0.4
100
99
51-98
50
Other
0.3
0.2
0.1
300
0
0
50
100
150
200
Response Time
250
300
Table 6: The Dictator Game
Imagine that you and another person (who you do not know) are to share $100. You need to decide how to split the
$100 between the two of you.
#187: How much will you give to the other person (and you keep the rest)?
#188: How much will you keep for yourself (with the rest going to the other person)?
The results of the two versions are remarkably similar. On average and in both versions, the
dictator gave $29 to the other person. In both versions, 24% of the subjects took the entire sum
for themselves and 11-12% gave the other player only $1. The fair division was chosen by 3640% of the subjects. In both versions, the fair division appears to be instinctive and indeed is
associated with the quickest response. Those who gave the other person less than $50 were
classified as contemplative. How to interpret the action of those gave more than $50 (3-5% of the
subjects) is uncertain: is it a case of generosity or of misunderstanding the problem? To be on the
safe side and given the small proportion of subjects in this category, these subjects were omitted
from the analysis.
Given the almost identical results of the two versions, the results are merged in the rest of the
analysis.
10 | P a g e 3.7. The Ultimatum Game (#23)
1
Percent
11%
13%
9%
49%
10%
7%
MRT
55s
53s
50s
40s
52s
47s
0.9
0.8
0.7
Frequencies
n=13957
0-1
2-39
40-49
50
51-60
61-100
0.6
0.5
0.4
0-1
2-39
40-49
50
51-60
61-100
0.3
0.2
0.1
0
0
50
100
150
200
Response Time
250
300
Table 7: The Ultimatum Game
Imagine that you and another person (who you do not know) are to share $100. You must make an
offer as to how to split the $100 between the two of you and he must either accept or reject your
offer. In the case that he rejects the offer, neither of you will get anything. What will your offer be? I
offer the following amount to the other person (and if he agrees I will get the remainder):
An equal division of the money seems to be the instinctive choice in the Ultimatum Game. Out
of 13,957 subjects, 49% chose to equally divide the money and their response time (40s) was
clearly the lowest. Thus, the choice of “50” is classified as instinctive. The MRT of the 11% of
subjects (the “victims of game theory”) who chose 0 or 1 is much higher (55s) as is the MRT of
subjects who chose a number within the range 2-49. Interestingly, 17% of the subjects chose a
number above 50. Such a choice might be the result of contemplative thinking or may be the
outcome of confusion between giving and taking. Unlike in the Dictator Game, these subjects are
a large proportion of the total and their response time does not indicate carelessness. Therefore,
they will be classified as contemplative.
3.8. The one-shot chain store game (#28)
1
Percent
47%
53%
MRT
84s
68s
0.9
0.8
0.7
Frequencies
n=7148
Enter
Not
0.6
0.5
0.4
0.3
0.2
Enter
Not to enter
0.1
0
0
50
100
150
200
Response Time
250
300
Table 8: The one-shot chain store game
In your neighborhood, there is one grocery store and one tailor. At the moment, the profits of the
grocery store owner are around $10K per month while the tailor's profits are only $4K per month. The
tailor asks your advice about whether to change his shop into a grocery store. He figures that if the
grocer does not respond aggressively to the new competition, each of them will earn about $6K per
month. On the other hand, if the grocer does respond aggressively and starts a price war, then the
earnings of each store will be reduced to about $2K per month. What is your advice to the tailor?
11 | P a g e Subjects (n=7148) were asked to play the role of the entrant in Selten (1978)’s one-shot chain
store game, which was presented verbally. The safe action, i.e. not to enter and to remain at the
lower but safer level of profit, seems to be the instinctive one. The MRT of the 53% of subjects
who chose the safe action (68s) was much lower than that of the 47% of subjects who chose the
risky action (84s). Thus, “Entry” is classified as the contemplative action and “No Entry” as the
instinctive one.
3.9. The Centipede Game (#33)
Percent
11%
10%
2%
9%
10%
58%
1
MRT
167s
108s
209s
174s
167s
139s
0.9
0.8
0.7
Frequencies
n=7111
1
2-95
96-98
99
100
101
0.6
0.5
0.4
1
2-95
96-98
99
100
101
0.3
0.2
0.1
0
0
50
100
150
200
Response Time
250
300
Table 9: The Centipede Game
You are playing the following "game" with an anonymous person. Each of the players has an
"account" with an initial balance of $0. At each stage, one of the players (in alternating order - you
start) has the right to stop the game. If it is your turn to stop the game and you choose not to your
account is debited by $1 and your opponent's is credited by $3. Each time your opponent has the
opportunity to stop the game and chooses not to, your account is credited by $3 and his is debited by
$1. The game lasts for 200 stages. If both players choose not to stop the game for 100 turns, the game
ends and each player receives the balance in his account (which is $200; check this in order to verify
that you understand the game). At which turn (between 1 and 100) do you plan to stop the game? (If
you plan not to stop the game at any point write 101).
Subjects (n=7111) were asked to play the role of the leading player in Rosenthal (1981)’s
Centipede Game with 100 turns. It was presented as a contemplative game (namely, a player is to
choose the turn in which he intends to stop the game; the number 101 stands for "never stop").
The choices in the range 96-100 had the highest response times and are classified as
contemplative. The choice of the Nash equilibrium strategy (“1”) is also classified as
contemplative. The instinctive action is “never stop” which is also the most popular choice
(58%). The choices in the range of 2-95 (10%) seem to be random and are also classified as
instinctive.
12 | P a g e 3.10. The "Stop or Pass Game" (#34)
In this game, each of 20 players in turn can either stop the game and receive $10 or pass. If all 20
players pass, each receives $11. Subjects were asked to imagine that they are the player who
starts the game.
1
Percent
61%
39%
MRT
44s
49s
0.9
0.8
0.7
Frequencies
n=6267
Stop
Pass
0.6
0.5
0.4
0.3
0.2
To stop the game
To pass the game on
0.1
0
0
50
100
150
200
Response Time
250
300
Table 10: The "Stop or Pass” Game
You are player number 1 among a group of 20 players participating in a 20-stage game. At stage t,
player t has to decide whether to stop the game or pass the game on to player t+1. If he stops the
game, he receives $10 while all other players receive nothing. If none of the 20 players stop the
game, then they all receive $11 each. Your choice is:
The response time results (n=6,267) support the intuition that stopping the game is the instinctive
action.
3.11. The Traveler’s Dilemma (#53)
1
Percent
21%
19%
5%
3%
8%
44%
MRT
97s
85s
116s
118s
102s
81s
0.9
0.8
0.7
Frequencies
n=15215
180
181-294
295
296-298
299
300
0.6
0.5
0.4
180
181-294
295
296-298
299
300
0.3
0.2
0.1
0
0
50
100
150
200
Response Time
250
300
Table 11: The Traveler’s Dilemma
Imagine you are one of the players in the following two-player game: Each of the players chooses an
amount between $180 and $300. Both players receive the lower amount. Five dollars are transferred from
the player who chose the larger amount to the player who chose the smaller one. In the case that the same
amount is chosen by both players, each receives that amount and no transfer is made. What amount would
you choose?
13 | P a g e In this version of Basu (1994)’s Traveler’s Dilemma Game, players announce a demand in the
range of 180-300 (n=15,215). If one player asks for a strictly lower amount than the other, he
receives an additional $5 at the expense of the other player. The instinctive action, chosen by
44%, is clearly 300 (MRT=81s). The choices in the range 181-294 (19%, MRT=85s) are also
classified as instinctive. The choice of alternatives in the range 295-299 (16%, MRT=109s) are
usually interpreted as an outcome of k-level reasoning and thus are classified as contemplative.
Another choice which is classified as contemplative is the unique Nash equilibrium choice of
180 (21%, MRT= 97s).
4. A new typology: the spectrum between Instinctive and
Contemplative
This section presents the core idea of the paper – a proposed new typology of players. The
typology is applicable (at most) in games like those discussed in Section 2, in which an intuitive
distinction is possible between actions chosen on the basis of contemplative considerations and
those that are the outcome of instinctive reasoning.
The typology proposed for such games defines a player’s type according to his tendency to
choose a contemplative action. In a formal model (whose development is in progress), an agent
would be characterized by (aside from more conventional characteristics) the probability of
choosing a contemplative action whenever he makes a decision.
Note that a player's type specifies the probability that he will choose a particular type of action
(i.e. instinctive or contemplative) rather than specifying the particular action that the player will
choose. This is in line with Arad and Rubinstein (2012) who found a correlation between the
actions of a player in the Colonel Blotto game and in the 11-20 money request game in this
sense, namely according to a classification of actions as instinctive/contemplative rather than a
classification on the level of strategies.
Given a set of games and a contemplative-instinctive division of strategies for each game, a
subject’s type is determined using the contemplative index (CI), which is defined as the
proportion of games in which he has chosen a contemplative action. A CI of 1 means that he has
always chosen a contemplative action and a CI of 0 means that he has always chosen an
instinctive strategy.
The typology is tested using the results from the 11 games described in Section 3. Only subjects
who played at least 7 of the 11 games are included. For each game, the CI of each subject is
calculated on the basis of the results for all the other games. Ideally, the correlation would be
calculated between CI and the probability that a player will play contemplatively in the eleventh
14 | P a g e game. However, since each player plays the eleventh game only once, this probability can be
estimated only by the frequency of the choice of contemplative actions within groups of players
with similar CI.
The following table presents 11 graphs, each containing ten points. The blue diamonds above x
on the horizontal axis indicate the proportion of subjects whose CI is between x and x+0.1 and
who chose a contemplative action. The red bars indicate the proportion of subjects whose CI is
between x and x+0.1. For example, the graph for game #15 is based on 1238 subjects who
responded to this problem and at least 6 of the other 10 games. The leftmost point on the graph
indicates that 34% of the 3% of subjects with CI in the range 0-0.1 chose the contemplative
action.
To further emphasize the relationship, each of the graphs also includes a linear regression line
where each point is weighted by the proportion of subjects in the corresponding range. Thus, for
example, the linear approximation of the ten points in game #15 is PC=36.6+14.7 CI.
That CI has some “predictive power” is quite clear from the graphs.
15 | P a g e Table 12: The
T
proportiion of conteemplative acctions as a ffunction of C
CI in the eleeven games. The
blue diam
monds abovee x on the ho
orizontal axiss indicate the proportionn of subjects whose CI iss
between x and x+0.1 and who ch
hose a contem
mplative actiion. The redd bars indicatte the proporrtion
of subjeccts whose CII is between x and x+0.1.
#15
5 ‐ Zero-Sum
m Game (n=
=1238) PC= 14.7 CI + 36.6
R2=0.38
=
#66 – Two-Conttest game (n
n=796) PC= 10.9 CI + 27.7
R2=0.32
=
PC= 1
16.5 CI + 41
1.3
#3 ‐ Rely
ying on ratioonality (n=11967) PC= 3
33.9 CI + 47
7.8
R2=0.9
90
R2=0.53
=
#195 ‐ Dictator Ga
ame (n=10112) PC= 3
33.6 CI + 53
3.5
16 | P a g e R2=0.4
40
#4 ‐ Su
uccessive Ellimination (n=2109)
(
PC= 23.8 CI + 18.4
#668 ‐ Hoteling
g’s Main Sttreet Game (n=1542) R2=0.6
65
#23 ‐‐ Ultimatum
m Game (n=
=1938) PC=
= 48.1 CI + 35.6
R2=0.88
=
#33 ‐ The
T Centipeede Game (n
n=2119) PC=
= 33.1 CI + 19.0
R2=0.78
=
#288 ‐ The one--shot chain sstore game (n=1619) PC= 36
6.2 CI + 38.9
##34 ‐ The "S
Stop or Passs” Game (n=1950) PC= 4
4.5 CI + 36.3
3
#53 ‐ Traveler’s
T
Dilemma
D
(n
n=1954) PC
C= 6.5 CI + 37.9
3
17 | P a g e R2=0
0.18
R2=0.8
88
R2=0.14
4
5. Results for additional games in light of the new typology
This section discusses several additional games in which actions cannot be intuitively classified
as contemplative or instinctive. The CI (calculated on the basis of the 11 basic games) is used to
provide an interpretation or even a “prediction” of the choices in these games.
5.1. Responders in the Ultimatum Game (#25 and #86)
Subjects were asked to play the game in the role of responder. They were randomly assigned to
respond to an offer of either $10 or $5 (out of $100).
#25 ($10)
Percent
62%
38%
MRT
25s
24s
n=4315
Yes
No
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Frequencies
Frequencies
n=7978
Yes
No
0.5
0.4
0.3
MRT
24s
23s
0.5
0.4
0.3
0.2
0.2
Yes
No
0.1
0
#86 ($5)
Percent
54%
46%
0
50
100
150
200
Response Time
250
Yes
No
0.1
300
0
0
50
100
150
200
Response Time
250
300
Table 13: Ultimatum Game: A Responder
You and someone you do not know are to share $100. He makes you an offer and you can either accept it or
reject it. If you reject it, neither of you will get anything.
#25: He offers you $10 (if you accept, he will get $90).
#86: He offers you $5 (if you accept, he will get $95).
Do you accept the offer?
About 62% out of the 7,978 subjects responded that they would accept an offer of $10 while
54% out of 4315 subjects responded that they would accept $5. In both cases, there is no
difference between the response time cdf’s of those who accepted the offer and those who
rejected it. That is, there is no indication that one of the two options is more instinctive than the
other. Nevertheless, the graphs in Table 14 show that CI is correlated with the acceptance rate. In
particular, the acceptance rate for the offer of $10 is 49% for CI up to 0.4 as compared to 68%
for CI above 0.4. Similarly, the acceptance rate for the offer of $5 is 47% among the subjects
with CI of up to 0.4 and 59% for CI greater than 0.4.
18 | P a g e Table 14: The
T
Ultimatu
um Game (rresponder):: The rate oof acceptancce as a functtion of CI #25 –
– Response to
t $10 (n=10
041) PC= 46.5 CI + 42.1
4
#86 ‐ R
Response to $5 (n=754)
R2=0
0.79
R2=0.46
6
PC= 25 .7 CI + 43.7
7
5.2. The Contributio
on Game (#
#79)
This is a standard fiv
ve-player Contribution Game
G
(see Occkenfeks andd Weimann (1999)). Eacch
player deecides how to
o allocate 10
0 tokens betw
ween a privaate fund (whhere a token iis worth $4 tto
the playeer himself) an
nd a public fund
f
(where a token is w
worth $2 to eeach of the fiive players).
1
Percent
38%
27%
13%
8%
15%
MRT
M
108s
101s
94s
109s
130s
0.9
0.8
0.7
Frequencies
n=8531
0
1-4
5
6-9
10
0.6
0.5
0.4
0
1-4
5
6-9
10
0.3
0.2
0.1
0
0
50
100
150
200
Respo
onse Time
250
0
300
Ta
able 15: The Contribution
n Game
Yo
ou are participaating in a gamee with four otheer players. In tthe game, eachh player gets 100 tokens.
Yo
ou have to deciide how to allo
ocate the 10 tok
kens between 2 different fundds: (1) Your peersonal fund:
forr each token yo
ou invest in you
ur personal fun
nd, you (and onnly you) will reeceive $4. (2) T
The group
fun
nd: for each tok
ken any playerr invests in the group fund, eaach of the 5 plaayers receives $2
(reegardless of how much they th
hemselves inveested in this fuund). Your deciision is how too divide the
10 tokens betweeen the 2 funds. You may mix up the investm
ment of your 100 tokens in anyy way you
wish.
I allocate
a
the folllowing amountt of tokens to th
he group fund::
RT seems too be
“No conttribution” is the most popular choicee (38% of thee 8531 subjeects). High R
correlated with sociaal behavior: the
t RT of thee 15% who ccontributed aall their tokeens to the puublic
19 | P a g e fund is ex
xceptionally
y high. The middle
m
option
n ($5 to the public fund)) attracts, as usual, a sizaable
proportio
on of the sub
bjects (13%) and its MRT
T is the loweest.
me reveals a different sttory.
Howeverr, the correlaation between
n CI and thee choices maade in the gam
CI is foun
nd to be corrrelated with the proportiion of subjeccts who chosse an extrem
me value (eithher 0
or 10); in
n other word
ds, having a higher
h
CI maakes it more likely that tthe subject w
will choose oone
of the tw
wo extreme allternatives. Players
P
who are more coontemplativee appear to bbe more decisive
about wh
hether or nott to contributte.
Table 16: The
Contriibution Gam
me: the prop
portion of cchoosers of 0 or 10 as a function off CI
#79
9 – The Contribution G
Game (n=13334) R 2=0.76
PC= 34.3 CI
I + 43.9
5.3. The Trust Gam
me (#133)
In this veersion of the Trust Gamee (Berg, Dick
khaut and M
McCabe (19995)), player A can transfeer up
to $10 to player B. The amount trransferred iss then tripledd and player B has to deccide how muuch
to give back to player A. Subjectts were asked
d to play thee game in thee role of playyer A.
1
Percent
28%
13%
18%
18%
9%
13%
MRT
M
96s
9
107s
93s
9
81s
8
84s
8
83s
8
0.9
0.8
0.7
Frequencies
n=6879
0
1
2-4
5
6-9
10
0.6
0.5
0.4
0
1
2-4
5
6-9
10
0.3
0.2
0.1
0
0
50
100
150
200
Re
esponse Time
250
300
Tablee 17: The Trust Game
Playerr A is given $10 and must decide how much
h to transfer too Player B ($x) and how muchh to keep ($10-x). Th
he two players don't know onee another and will
w not meet. A
After Player A has made his decision,
Playerr B is informed
d of x. The amo
ount x is then tripled,
t
and Plaayer B must deecide how much (out of 3x) too
keep and
a how much to give to Play
yer A. The amo
ount Player B ggives will be aadded to what P
Player A kept tto
himseelf from the original $10. If yo
ou were Playerr A, how muchh of the $10 woould you give tto Player B?
20 | P a g e Once agaain, the midd
dle choice ($
$5) is populaar (18% of thhe 6,879 subj
bjects) and thhe MRT of thhose
who chosse it is the lo
owest (81s). The MRT (97s)
(
of thosse who chosee to transfer less than $5
(59% of the
t subjects)) is much hig
gher than forr those who transferred m
more than $5 (83s).
Thinking
g about a cho
oice for a lon
nger time app
parently makkes the subjeect less trusttful.
The CI an
nalysis yield
ds even stron
nger results (see
(
Table 18). Thus, traansferring at least $5
becomes less popularr as CI increeases and thee proportion of subjects w
who expresss complete
(
g 0) increases with CI.
mistrust (transferring
Table 18: The
T
Trust Game:
G
propo
ortion of sub
bjects who cchose 5-10 aand 0 as a fu
unction of C
CI #133 –Trustt game (a ch
hoice of 5-10
0) (n=1035)
PC= -34.7 CI + 51.5
R2=0
0.66
#1133 –Trust g
game (a chooice of 0) (n=
=1035) PC= 39 .1 CI + 16.5
5
R2=0.77
7
5.4. Chiccken (#77).
Chicken is a game with
w no discerrnible correlations betweeen behaviorr and either R
RT or CI. Thhe
game waas presented to subjects as
a a 2×2 mattrix game, w
without any bbackground sstory. The
populatio
on of 8837 su
ubjects was split almost evenly betw
ween the twoo actions. Thhe response ttime
cdf’s are very similarr and the CI is not correllated with thhe choices m
made in the gaame.
21 | P a g e 1
Percent
P
51%
49%
MR
RT
51ss
48ss
0.9
0.8
0.7
Frequencies
n=8837
A
B
0.6
0.5
0.4
0.3
0.2
A
B
0.1
0
0
50
100
15
50
200
Respons
se Time
250
300
Tablle 19: The Chiicken Game
You are player 1 in
n a two-person game with the following monnetary payoff m
matrix:
Player 2
A
B
A
3,3
2,4
2
Play
yer 1
B
4,2
1,1
1
Whaat will you play
y?
Table 20: The
T
Chicken
n Game: pro
oportion of subjects wh
ho chose A aas a function
n of CI #77 – Ch
hicken (n=1 329) PC=
P
-0.9 CI + 54.1
22 | P a g e R 2=0.002
6. The Global Index
The typology proposed above classifies subjects using observations of their past behavior.
Rubinstein (2013) proposed an alternative index which classifies subjects according to their
response time, i.e. fast or slow. The calculation of the index is as follows: For each subject and
each problem in the dataset (which was responded to by at least 600 subjects), the subject’s
“Local rank” is defined as the fraction of subjects who answered that problem faster than he did.
A “Global Index” (GI), which is the median of the local rankings, is calculated for each subject
who answered at least 9 problems (the median number of answers for a participant who received
a global rank is 15). The GI was calculated for about half of the subjects. The distribution of
global rankings is presented in Rubinstein (2013) and shows that the distribution of GI is much
more dispersed than if the local rankings were distributed uniformly. This implies the existence
of a significant correlation between the local rankings of the same subject in different problems.
The correlation between local and global rankings is almost always between 0.5 and 0.7.
The connection between GI and the tendency to choose contemplative actions is demonstrated in
Table 21. Once again, each graph refers to one of the 11 games. Each point refers to a decile in
the GI ladder. The k’th blue diamond relates to the subjects in the k’th decile (from the bottom).
It indicates the median GI in the corresponding decile and the proportion of subjects in the decile
who chose a contemplative action.
The positive relationship between GI and contemplative behavior is evident from the graphs,
although the positive trend is less pronounced than was the case for CI. (The average regression
coefficient is 19 as compared to 24 in the corresponding regressions for CI.)
23 | P a g e Table 21: The
T
proportion of conteemplative acctions as a ffunction of G
GI in eleven
n games. Th
he
k’th bluee diamond relates
r
to th
he subjects in
i the k’th d
decile (from
m the bottom
m). It indicattes
the mediian GI in th
he correspon
nding decilee and the prroportion off subjects in
n the decile w
who
chose a contemplati
c
ive action.
#15 – Zerro-Sum (n=1238) #68 ‐ Hotteling’s (n=1
1542) PC= 26.6 GI
G + 29.4
R2=0.65 #3 ‐ Ratiionality (n=1
1967) PC= 15.2 GI
G + 52.7
PC= 29.1 GI + 32.7
R =0.55 2
PC= 21.1 GI + 17.4
R =0.39 2
PC= 27.0 GI + 39.0
24 | P a g e R =0.68 2
R =0.94 2
#53 ‐ Tra
aveler (n=1 954) R2=0.12 P
PC= 14.7 GI
I + 59.0
R2=0.75 R 2=0.62 R =0.36 2
#33 ‐ The C
Centipede (n
n=2119) P
PC= 18.9 GI
I + 22.9
PC= 17.2 GI + 31.8
P
PC= 23.4 GI
I + 20.5
##195 ‐ Dictattor Game (n
n=1012) #28 ‐ The ch
hain store (n
n=1619) #34 ‐ "Stop
p or Pass” (n
n=1950) PC= 6.8 GI
I + 34.8
R 2=0.66 #4 ‐ Elimination (n=2
2109) #23 ‐ Ultiimatum (n=
=1938) PC= 13.3 GI
G + 46.7
#66 – Two
o-Contest (n
n=796) R =0.72 2
7. Discussion
A novel typology is proposed for players of games in which a clear distinction can be made
between instinctive and contemplative strategies. According to this typology, a player is
characterized by his tendency to choose a contemplative action (as opposed to an instinctive
action).
The proposed typology is a hybrid of two approaches to classifying agents: one which uses
observed behavior in previous games as a predictor of behavior in other games and one which
uses response time in previous games to predict a player’s behavior in other games. Response
time data is used to confirm the intuitive partition of actions into contemplative (long RT) and
instinctive (short RT). Correlations are calculated in the domain of “classes of actions” rather
than the domain of “individual actions”. A highly contemplative type is an agent who tends to
choose contemplative actions more often than instinctive actions.
Another typology based on the Global Index is also presented. It classifies subjects as fast or
slow according to their median position on the response time ladder. Overall, the correlations
between GI and the choices made appear to be somewhat weaker than for CI.
Non-objectively I find the results interesting. However, as the cliché goes: “Much more work is
needed…“
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