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Tsukuba Economics Working Papers
No. 2009-003

Born Under a Lucky Star?
by
Nobuyuki Hanaki, Alan Kirman and Matteo Marsili
March 2009

UNIVERSITY OF TSUKUBA
Department of Economics
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
JAPANBorn Under a Lucky Star?∗Nobuyuki Hanaki†

Alan Kirman‡

Matteo Marsili§

March 6, 2009

“I am a great believer in luck,and I find the harder I work,
the more I have of it”
Thomas Jefferson

Abstract
This paper suggests that people can learn to behave in a way which makes
them unlucky or lucky. Learning from experience will lead them to makechoices which may lead to “luckier” outcomes than others. By so doing they
may reinforce the choices of those who find themselves with unlucky outcomes.
In this situation, people have reasonably learned to behave as they do andtheir behaviour is consistent with their experience. The lucky ones were not“born under a lucky star” they learned to be lucky.

Keywords: Learning, Search
JEL code: D83
This work was partially supported by EU project 516446 ComplexMarkets under FP6-2003NEST-PATH-1 and by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grants-in-Aid for Young Scientists (B), No. 19730164. We thank the participants in theseminars at which this paper has been presented, in Boston, Paris, Warwick and Cagliari, for
helpful comments. The usual disclaimer applies.
†
Department of Economics, Graduate School of Humanities and Social Sciences, University of Tsukuba.
1-1-1, Tennodai, Tsukuba, Ibaraki, 305-8573, JAPAN. E-Mail:
hanaki@dpipe.tsukuba.ac.jp‡
GREQAM, EHESS, IUF, 2 Rue de la Vieille Charité, 13002 Marseille, France. E-Mail:alan.KIRMAN@univmed.fr.
§
Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34014, Italy.
Email: marsili@ictp.it
∗

1

1

Introduction

There are many attitudes to success. For some it is jealousy, based on the belief thatthe successful are just lucky and that they do not merit what they have obtained,
for others it is admiration for the effort or skill that must have led to the success.
What is interesting is that there does not seem to be a clear and commonly accepted
definition of luck. The Oxford Dictionary provides us with the following definition:“success or failure apparently brought by chance rather than by one’s own actions.”
The idea that “luck” is exogenous to one’s own actions has led to a debate on
the importance of “luck” versus “action” in determining success. Discussions suchas “luck” versus “skill” in the management and finance literature (e.g., Hartzmark,1991; Gompers et al., 2006; Cuthbertson et al., 2008), and “luck” versus “policy”
in the macroeconomic literature (e.g., Ahmed et al., 2004; Leung et al., 2004) are a
few examples.
People’s perception of the importance of “luck” versus “effort” in determining
ones’ success can have a strong effect on the degree of government intervention ina society (Piketty, 1995; Alesina and Angeletos, 2005) or severity of punishment
for criminals (Di Tella and Dubra, 2008). These authors argue that in a society
where people believe that individual efforts determine success, a lower degree ofgovernment intervention or a severer punishment for criminals will be chosen. On
the other hand, the degree of government intervention will be higher in a society
where people believe that luck, not individual efforts, determines success.
Regardless of the belief in a society about the importance of luck in determining
one’s own success, we have many expressions such as “fortune favours the brave”
or “nothing venture nothing gain” which suggest that behaviour and attitudes cancontribute to apparently lucky favourable outcomes.
Take the simple example of car insurance. Here two factors are in play. Firstly,
there is the idea that certain events are due purely to “bad luck” and that the consequences of these should be mutualised and then there is the problem of adverse
selection. Those who insure themselves are more likely to be those to whom “unlucky” events may happen. Bad drivers do not have the same probabilities of an
accident as good drivers almost by definition. Insurance policies with their “bonus”
and “malus” are structured to separate out these two problems.
But this means that, since it is possible to improve one’s driving by learning,
one can learn to become “lucky” or, put alternatively, to experience fewer negative
shocks than others. An alternative view, that of a psychologist Richard Wiseman(2003), suggests that “lucky” people are more aware of the opportunities that arise2and hence more likely to profit from them. Thus it is not enough to study theprocess of arrival of “good” or “bad” luck but rather the attitude of the actors
themselves. Others have argued that more optimistic people tend to be luckier andsome psychologists such as Martin Seligman (1991) suggest that one can learn to be
optimistic.
It is actually quite difficult to cite examples of “pure luck”. Winning a large
sum in a National Lottery might seem like an obvious example. However, you canlearn how to increase the expected value of the outcome, at least in some countries,
by studying the numbers that people choose. Some are very popular and some are
hardly chosen at all. Therefore, armed with this information, one should choose an
unpopular number since, if one wins, one has to share her gains with fewer people.
This remark points to another important feature of certain lucky outcomes whichis that they are part of a constant sum game. What one individual gains deprives theothers of that opportunity. Thus, good luck for some implies, in this framework, badluck for others. This means that if one person learns how to seize lucky occasions
he may deprive others and thereby make them “unlucky”.
What we will suggest in this paper is that people can, in some cases, learn tobehave in a way which makes them unlucky or lucky. Learning from experience willlead them to make choices which may lead to “luckier” outcomes than others. By
so doing they may reinforce the choices of those who find themselves with unlucky
outcomes. Our simple example is based on the everyday experience of one of us inparking his car. There are many parking spots in the small streets leading to theVieille Charité in Marseille, where the group to which one of us belongs, is based.
Having taken a spot, one then walks the rest of the way and this takes time andeffort. What one observes, is that some people are systematically parked close to theoffice while others are always far away. Excluding ex-ante heterogeneity of agentsin their access to the parking, the natural question is: are the former just lucky or
is something else at work? We wish to suggest that people have reasonably learned
to behave as they do and that their behavior is consistent with their experience.
The lucky ones were not “born under a lucky star”, they learned to be lucky.1 The
second important issue concerns those aspects in the environment of agents whichfavors asymmetric outcomes with individuals being persistently lucky (or unlucky).
We find that such state of affairs to prevails depending both on the way agents learn,
Note that, in this paper, we are not trying to explain the reason for which people tend to
attribute success to their own skills and failures to bad luck. Consideration for such “self-serving”
bias can be found, for example, in Compte and Postlewaite (2004); van den Steen (2002) and theliterature cited therein.
13

on payoffs and on the timing of the search process. Specifically, an high degree ofsynchronization in the access to parking, with most agents going to and leaving workat roughly the same time (in the morning and in the afternoon, respectively), favors
asymmetric outcomes whereas asynchronous access to resources favors symmetricoutcomes. Interestingly, increasing the cost agents incur if they fail to find a parking,favors asymmetric outcomes only in an intermediate range.
Although the model will be presented with a particular example of parking along
the one way street, the same idea can be applied to other situations in which agents
search for better deals. For example, in a market place, buyers search for cheaper
prices by visiting several sellers.2 What buyers do not know in this case is the prices
offered by various sellers in the market. In the standard model of search, buyers
have some beliefs about the underlying distribution of prices offered by the sellers
in the market, and decide to accept the proposed price by comparing the cost andthe expected benefit of continuing their search for better offers.
There is a literature on the optimal strategy to adopt when one is faced with aseries of prospects but with no possibility of revisiting previous candidates. This isstudied by Krishnan (2006) who likens the problem to that of choosing an exit froma highway. Once one has taken a particular exit one cannot, in general, go back toprevious exits nor profit from later ones. However, the fact that the behaviour ofothers influences the choices available is not taken into account.
Our problem also is related to the famous “secretary problem” originally posed
by Gleason in 1955 according to Gilbert and Mosteller (1966). In this problem
surveyed by Freeman (1983) and Chow et al. (1991), one is faced with a string ofsecretaries and has to choose one. In the classical version the number of secretaries
is finite and if one gets to the last candidate that candidate has to be selected. The
problem is when to stop and choose the current candidate. As in our case there isno going back. Two features are different from those of our problem. Firstly, as
in the previous case the interference from other secretary hirers is not taken into
account. Secondly, in our problem the value of each successive slot is increasing but
only if it is free. Otherwise it provides no value to the person searching. Gilbert
and Mosteller (1966) did, in fact, treat a strategic version of the problem since theselector is faced with an opponent who can decide in which order to present the
Literature on learning by buyers in the market mostly focuses on buyers learning which seller
to visit. See, among others, Weisbuch et al. (2000); Hopkins (2007), which also reports evidenceof consistent heterogeneity in prices across buyers. Of course, it is not only buyers who learn inthe market but also sellers learn about the best price to charge. See Hopkins and Seymour (2002)
for an interesting analysis for such cases.
2

4

candidates.
Steblay et al. (2001) look at the problem of a police line-up in which the suspects
are presented sequentially but the person who has to identify the criminal has to
identify him on the spot and cannot go back to the previous suspects. Again, there
is no competition for the suspects amongst those identifying.
Lee et al. (2004) examine the capacity of people to choose in this sort of problem
in an experimental setting. Their interest was in the psychological aspect of the
choice making which is, of course, related to our idea that luck is not random but
due to other factors.
The question the standard search literature does not address, in general, is how
buyers form their beliefs about the distribution of opportunities (see, however, Rothschild, 1974). Our working hypothesis is that agents learn what to accept throughtrials and errors, although they may not necessarily learn the underlying distribution of opportunities. In other words, in our paper, the agents are not learning tosolve this problem, which is why they can converge to different solutions in spite of
the fact that they face the same problem. Notice, furthermore, that in our context
the underlying distribution of the opportunities available to an agent is determined
endogenously, as it depends on what other agents have learned to accept.
While our emphasis is on learning, we shall first propose an analysis of somestatic results, for large populations, in two extreme cases. These two cases bringpolar predictions on the emergence of lucky and unlucky individuals. We shall thenproceed to analyze the learning dynamics. However, a simple theoretical analysis
can be done only in the case of two agents. The general case of a large populationof agents will be analyzed through numerical simulations. What we find, confirms
both the qualitative features disclosed by the analysis of learning in the case of two
agent, and the polar predictions of our static analysis.

2

The Model

Consider a hypothetical city where there is a one way street leading to the city center.
This is the unique street in the city where people can park their cars. Suppose there
are S parking spots that are distributed evenly along the street. Parking spots areindexed by their distance from the city center, so that spot s ∈ {1, 2, ..., S} is at
distance s − 1 from the center.
There are N ≥ S agents living around the city.3 At the beginning of each period3

The case of N < S is equivalent to considering the case with N parking spots although the5t ∈ {0, 1, 2, ..., }, each agent can be either in the city, already parked, or outside thecity. If an agent is outside the city, there is a probability pe that she comes into thecity and looks for a parking spot along the one way street. When an agent looks fora parking spot, she chooses a strategy k ∈ {1, .., S} which means that she will parkat the first empty spot, s, that is closer to the center than spot k + 1 (i.e, s ≤ k).
When she parks at spot s, she receives the payoff π(s). The payoff is normalized
between zero (π(S) = 0) and one (i.e. π(1) = 1), and is decreasing in s, i.e, thecloser the spot is to the center, the higher the payoff is. As a specific case, we shallconsider the payoff for parking at spot s as being given by
π(s) = 1 −

!

s−1
S−1

"α

(1)

where α > 0 is a parameter that determines the curvature of the payoff function.
If she did (could) not park before reaching the city center, she has to park at
a pay parking lot. Because of the fee this entails, she receives payoff −L (whereL ≥ 0) when she fails to park before the city center, i.e., the spot 1.
After all the players who looked for a parking spot made their decisions (somehave parked along the street and others may have failed to park before the city
center and parked in pay parking lots) and received their payoffs, each parked agent,
including the ones who found a spot in this period, leaves the city with probability
pl . If an agent leaves the city, she will be outside the city in the beginning of thenext period, while if she does not leave the city, she enters next period “in the city
already parked.”3

Equilibrium

In an equilibrium setting, the existence of lucky and unlucky individuals translates
into the presence of asymmetric Nash equilibria where agents play different strategies, some leading to better outcomes than others.
Our purpose here is not to provide a thorough characterization of Nash equilibria
in this game but rather to show that asymmetric equilibria, where some agents areluckier than others, can be sustained as equilibria for particular values of parameters
where symmetric equilibria do not exist. Likewise, symmetric equilibria where agents
adopt the same strategy, happen to be stable in a different region of parameter space.
We first consider the case pe , pl & 1/N , where the distinction between different
payoffs have to be normalized accordingly.

6periods becomes immaterial, and then the case pe = pl = 1, where each period
can be considered as a different game. In the former case (pe , pl & 1/N ), we findthat symmetric equilibria prevail in the sense that they exist in a broad region ofparameter space whereas only in a rather limited region asymmetric equilibria exist.
In the latter case (pe = pl = 1), we shall see that the converse is true. Symmetric
Nash equilibria only exist for rather specific choices of parameters. The intermediate
region between these two extremes should reveal a qualitative change between thetwo polar behaviors. This will be explored with our numerical simulations of learning
dynamics later in the paper.

3.1

The case pe , pl & 1/N

Let us consider case where S = N and the probability of agents entering and leaving
the city in any round is vanishingly small. More precisely, we take pe , pl → 0 but wekeep ratio pe /pl = η finite. This limit allows us to treat the turnover of agents in thecity as a continuous time process, where agents leave the city at unit rate and enter
the city at rate η.4 Take a particular agent i. After a transient time, her probability
to be in the city will converge to a stationary value. The latter should be suchthat the probability per unit time of her going into town, given that she was not in
town, equals the probability per unit time of her leaving the city. This condition –generally referred to as detailed balance – between agents entering and leaving thecity in a time interval, implies that any particular agent, in the stationary state, will
be in the city with probability
1−h=

η
1+η

(2)

Since arrival and departures are independent events, the number C of cars currently
parked, in the stationary state, has a binomial distribution! "N
P {C = c} =(1 − h)c hN −cc(3)Agents adopt threshold strategies labeled by an integer k: They will go to position k and then park in the first free parking s ≤ k which they find. If there is nofree parking slot, they incur the cost L. Call σi,k a mixed strategy for agent i, which
In this section, time t will be considered a continuous variable. One period of the gamecorresponds to a vanishingly small time interval dt = pl . In this time variable, the probability thatagent i enters (leaves) the city in time interval [t, t + dt) is ηdt (dt) for all t and for all agents i outof (in) the city.
47

is the probability that i picks strategy k. We want to investigate Nash equilibriaof this N player game. Since equilibria are not the main subject of this paper, we
relegate the derivation to the appendix, for the sake of clarity of exposition. Here
we shall confine our discussion to the main results and a basic intuition.
In order to discuss asymmetric Nash equilibria, consider the set of pure strategies{σi,k } = δk,ki ,(k1 , . . . , kN ) a permutation of (1, . . . , N )(4)where δi,j = 0 if i (= j and δi,i = 1. We find that:
Proposition 1: The strategy profile (4) is a Nash equilibrium, provided:
"k
N !
1 #
1
L≥
1+
π (k) .
1 + η k=1η(5)This implies that, for any finite L and η, the strategy profile in Eq. (4) is a Nashequilibrium only for N small enough. For finite η and large N , the right hand sideof Eq. (5) is exponentially large with N , i.e. a huge cost L is required in order forthe asymmetric Nash Equilibrium to exists. Conversely, when η ∼ N is also verylarge, the condition (4) is satisfied even for relatively small L.
The intuition behind this result is simple: In order for the asymmetric state
to be stable it must be the case that the worst off agent, who is parking at spot
N , has no incentive to deviate, i.e., her expected payoff from deviating to otherstrategies be negative. Only if there is no free spot, which occurs with exponentiallysmall probability (1 − h)N −1 , will she have to pay the cost L. So the penalty isirrelevant unless L(1 − h)N −1 is of order one. A formal derivation, which is given in
the appendix, leads to the specific prediction of Proposition 1.
Let us now consider a symmetric Nash equilibrium where every agent uses thesame strategy σk . In general, a mixed strategy is a distribution on k conditional on
the information available to agent i. This in particular includes the information on
the spots in which she has parked in the past. For the present discussion, we assume
agents to be naı̈ve, i.e., to disregard this latter piece of information. Accordingly, we
shall speak of equilibria with naı̈ve agents. This can be justified for large population
sizes (N * 1), in view of the random arrival of agents in the city.
Proposition 2: A symmetric equilibrium with naı̈ve agents exists in pure strategies,
i.e. σk = δk,k∗ for any value of η and L.
As shown in the appendix, in the limit as N becomes large, this equilibrium has
8the following properties:• the fraction z ∗ = k ∗ /N of potentially occupied parking slots is less than thefraction 1 − h of agents wishing to park. In other words, a positive fraction
1 − h − z ∗ of agents have to pay the cost L• the threshold z ∗ = k ∗ /N increases with L and with η (i.e. with the average
number of agents trying to park).
• the probability es that a parking spot is free decreases with s, i.e., the more
profitable a parking spot is, the more likely it is that it is unoccupied.
This equilibrium is not particularly efficient, first because a finite fraction of theagents have to pay the cost L, secondly because the best parking slots are thosewhich are more likely to be unoccupied. Indeed all parking slots in the asymmetricequilibrium are occupied with the same probability and the average payoff of agentsis higher than in the symmetric equilibrium. Hence, the asymmetric equilibrium
is more efficient than the symmetric one, both in terms of better exploitation of
resources and of the average payoff of the agents.
In summary, the analysis of the (naı̈ve) Nash equilibria of the N players game,for pe , pl & 1, suggests that both symmetric and asymmetric equilibria can arise.
The latter, when N is large, arises either when the cost L is extraordinarily high,or when the number of free spots is small.

3.2

The case pe = pl = 1

Next we turn to the case where all the parked agents leave the city at the end of
the period (pl = 1) and all the agents who are outside the city at the beginning of
the period come into the city (pe = 1). To make our analysis simple, we consider
the situation where the number of agents wishing to park in the city are the sameas the number of spots available, N = S.
We again consider two outcomes, the symmetric one and the completely asymmetric one, and see whether they can be Nash equilibria of the game.5 The symmetric outcome is where everyone chooses the same strategy (i.e., k i = k for all i) while
the completely asymmetric outcome is one where everyone chooses different strategies (i.e., without loss of generality, k i = i for all i). The results can be summarized
in the following two propositions:5

We confine our discussion to Nash equilibria in pure strategies.

9

Proposition 3: (i) Symmetric outcomes, where everyone chooses the same strategy,
cannot be a Nash equilibrium of the game when N − (N − 1)1+α < 0. (ii) If 0 ≤NL ≤ (N −1)
1+α − 1, then, the symmetric outcome where everyone chooses strategy 1,i.e, k i = 1 for all i is a Nash equilibrium of the game.
Proposition 4: The completely asymmetric outcome where everyone chooses a
different strategy is a Nash equilibrium of the game when$%1
L > max (N − j + 1)π(N − j + 1 − l) .
j
(l
+
1)l
l=1
N −j

#

(6)

It is worth remarking that the stability of the symmetric Nash equilibrium, forlarge N , requires that α and L lie within a very narrow range. With a little algebra,
the condition in Proposition 3, can be rewritten as α < − log(1−1/N )/ log(N −1) +1/(N log N ). When this condition is satisfied, the value of L also needs to be veryNsmall. Indeed we have L < (N −1)
1+α − 1 < 1/(N − 1), which implies that both α
and L should vanish as N → ∞, for the symmetric Nash equilibrium to exist. We
will now prove the two propositions.
To check whether the symmetric outcomes where k i = k for all i, that is when all
individuals choose the same strategy, can be a Nash equilibrium, we need to check
whether a unilateral deviation to k̂ (= k is profitable.
Since, each period, agents arrive at the street in a random order, the expected
payoff for an agent when everyone is choosing strategy k depends on how many
agents have already arrived at the street before her. For example, if i is the firstto arrive which happens with probability 1/N , then she will park at the spot k andobtain π(k). If she is the second, which again happens with probability 1/N , thenshe will park at spot k − 1 and obtain π(k − 1), and so on. That is, if she is thej(≤ k)-th agent to arrive at the street in that period, she will park at spot k −(j −1)
and obtain π(k − (j − 1)). But if there are more than k agents before her (i.e., sheis the j(> k)-th agent to arrive), then, she will not be able to park before the center
and will obtain −L.
Therefore, the expected payoff for an agent when everyone is choosing strategy
k is
$ k
%
#
1
E(Πi (k|k −i = k)) =π(l) − (N − k)L
(7)
N l=1
Now consider the expected payoff from a unilateral deviation. An agent, say i,can deviate and choose strategy k̂ = k + 1 or k̂ < k, in particular, k̂ = 1. The
10

expected payoff for agent i who deviate to k̂ = k + 1 while everyone else is choosing
strategy k can be easily obtained because i can park at spot k + 1 for sure. That is,E(Πi (k + 1|k −i = k)) = π(k + 1)

(8)

Now the expected payoff for agent i who deviates to k̂ = 1 when all the otheragents are choosing strategy k ≥ 2 can be derived as
E(Πi (1|k −i = k, k ≥ 2)) =1(kπ(1) − (N − k)L)N

(9)Intuitively, when all the other players are choosing strategy k, unless there are kother agents in the street before agent i, i can park at spot 1. If there are morethan k agents before i, i fails to park and obtains −L.
Now, one can check if k i = k for all i can be an equilibrium. We do this byconsidering first k = 1 and then k > 1.
When all the agents choose k = 1, unilateral deviation to k̂ = 2 is profitable if
E(Πi (1|k −i = 1)) < E(Πi (2|k −i = 1)). That is, from equations (7) and (8),1(π(1) − (N − 1)L) < π(2)NorL>
With π(s) = 1 −& s−1 'αS−11(π(1) − N π(2))
N −1

(10)

and N = S, we haveL>N
−1
(N − 1)1+α(11)

But note that if N −(N −1)1+α < 0, this condition is always satisfied because L ≥ 0.
Thus, k i = 1 for all i cannot be a Nash equilibrium when N − (N − 1)1+α < 0.
Let us now move to the case where k > 1. From equations (7) and (9), unilateraldeviation to k̂ = 1 is profitable if
1
N

$ k
#
l=1

π(l) − (N − k)L%<1(kπ(1) − (N − k)L)N

that is, ifk#π(l) < kπ(1)

l=1

11

(12)

which is always true for all k > 1 given the definition of π(s). Thus k i = k(> 1) for
all i cannot be a Nash equilibrium.
From equations (11) and (12), the symmetric outcomes cannot be a Nash equilibN
rium of the game if N − (N − 1)1+α < 0. On the other hand, if 0 ≤ L ≤ (N −1)
1+α − 1,ithen k = 1 for all i is a Nash equilibrium of the game, as stated in Proposition 3.
For example, in the case of a linear payoff (α = 1.0), a symmetric outcome is not aNash equilibrium for N > 2.
Now let us consider whether the completely asymmetric outcome can be a Nashequilibrium. Without loss of generality, let us consider an outcome such that agent
i chooses strategy i, i.e, k i = i for all i. Since the payoff is the lowest for i = N
in this particular case, agent N has the highest incentive to deviate. We consider
the possible payoffs that i = N could obtain by deviating from k N = N to anotherstrategy k N = k < N .
The expected payoff for agent N of a deviation to strategy k N = N − 1 whileother agents are using k i = i for all i < N , ΠN (N − 1|k i = i, ∀i (= N ), can becomputed for generic N :E(ΠN (N − 1|k i = i, ∀i (= N )) =

N
−1
#

1
L
π(N − l) −(l + 1)l
N
l=1

(13)

In general, the expected payoff from a deviation to strategy k N = N − j(j ≥ 1) isgiven byNiE(Π (N − j|k = i, ∀i (= N )) =

N −j

#

1
L
π(N − j + 1 − l) −(l + 1)lN −j+1l=1(14)

Since π(N ) = 0, the condition for k i = i for all i to be an equilibrium is, therefore,
given by Eq. (6) as stated in Proposition 4. Note that the critical L increases with
N . The larger is the number of agents (and parking spots), the higher is the costrequired to sustain this equilibrium.6
Summarizing, when everyone searches for parking spots every period, i.e, pe =pl = 1, we find that symmetric Nash equilibria exist only for α and L being verysmall. We have also shown that unless the loss of not parking before the center, L,
is large enough,7 the completely asymmetric outcome is not a Nash equilibrium.
It should be noted that deviating to N − 1 does not necessarily generates the highest expected
payoff for player N .
7
In the case of a linear payoff (α = 1.0), the critical L for having the completely asymmetric
Nash equilibrium is larger than the maximum gain from trying to parking at free spots along the
612

For intermediate values of α and L we do not have a characterization of Nashequilibria for general N . For α = 1 and L = 0 we found the following Nash equilibria
for small N

N = 2 : k = {1, 1}
N = 3 : k = {1, 1, 2}
N = 4 : k = {1, 1, 2, 2}
N = 5 : k = {1, 1, 2, 2, 3}
N = 6 : k = {1, 1, 2, 2, 3, 4}...
which suggests that as N increases Nash equilibria exhibit an increasing degree ofasymmetry. For L > 0 we expect an even higher degree of asymmetry.

4

Learning dynamics

What we are interested in this paper, however, is not Nash equilibria per se. We areinterested in whether agents learn to behave in such a way that some agents choose
strategies with small k (lucky ones) while others chooses strategies with large k(unlucky ones). In order to address this point, we need to introduce the learning
processes.
As stated above, every time agent i searches for a parking spot, she chooses a
strategy k that involves her parking at the first empty spot that is no worse than (asclose to the center as) spot k (i.e., s ≤ k). The choice of strategy in period t depends
on her past experiences from choosing (and not choosing) various strategies. The
past experiences are summarized by agent i’s attractions to each strategy, Aik (t), at
the beginning of the period.
The probability with which she chooses strategy k in period t isexp(λAik (t))
Pki (t) = (Sij=1 exp(λAj (t))(15)

where λ is a parameter of the model that determines the sensitivity of strategy choice
to attractions. If λ = 0, all the strategies are equally likely to be chosen regardlessstreet, L ≥ π(1) − π(S) = 1 for all N ≥ 2.

13

of their attractions. As λ becomes larger, the strategies with higher attractions
become disproportionally more likely to be chosen and, if λ = ∞, the strategy withthe highest attraction will be chosen with probability one. As shown by Weisbuch etal. (2000), λ controls the balance between exploration and exploitation. The logistictransformation introduced here is common in the literature on learning (see, forexample, Erev and Roth, 1998; Camerer, 2003; Brock and Hommes, 1997, 1998).8
We assume that all the agents have the same attractions for all the strategies atthe beginning of period zero, and that this is equal to the average payoff of parking(
along the street, i.e., Aik (0) = S1 Ss=1 π(s) for all k and i.
After searching for a parking spot in period t, the attraction for strategy k evolves
as follows:
Aik (t + 1) = ωAik (t) + (1 − ω)Rki (t)(16)
where ω is the weight put on the past value of attraction. Here the reinforcement
Rki (t) is equal to the payoff i has received in period t for all the parking spots where
agent i has actively searched for a spot, i.e.,Rki (t) =)π[si (t)] if0si (t) ≤ k ≤ k i (t)
else

(17)

Here k i (t) is the strategy i has chosen in t and si (t) is the spot i has parked in that
period, i.e., the first empty spot i found such that s ≤ k i (t). We set the convention
that si (t) = 0 if i failed to find an empty spot, and we set accordingly π(0) = −L.
Outside the searched interval, i.e. for k (∈ [si (t), k i (t)], we set Rki (t) = 0, and theattractions depreciate. Note that attractions are not updated and remain constantwhen the agent i does not search.
The fact that agents do not update attractions for strategies outside the interval
between where they start searching and where they actually park is reasonable ina volatile context such as the one we are interested here. Opportunities may havea short lifetime and be picked up by other agents. Counterfactually updating theattractions of strategies which were not really played may be unfeasible and/or
unrealistic.
Erev and Roth (1998); Camerer (2003) use the logistic formulation to better explain the behavior of experimental subjects in the laboratory experiments. Motivated by their experimental
results, McKelvey and Palfrey (1995) use this idea to develop the ”Quantal Response Equilibrium”,
which they consider to be a better solution concept because it allows for noisy action choices byplayers. Brock and Hommes (1997, 1998) use this logistic function in models where players learn
about performance of various price forecasting strategies in market setting and decide which strategies to choose.
814

In the model considered here, all the agents are ex ante identical. In particular,
we assume there is no heterogeneity in risk preferences among agents in order tounderstand the role of learning in generating heterogeneous behaviors. However, one
interpretation of our results is that agents learn which level of risk aversion to have.
Thus the “unlucky” agents have learned to be more risk averse than the “lucky”
ones.
As discussed in the introduction, the model has a wider applicability than theparticular example of parking along the one way street. In general, it describes asituation where agents learn what to accept, while they form their beliefs about the
distribution of opportunities. The underlying distribution of the available opportunities, in its turn, is determined endogenously, as it depends on what other agents
have learned to accept.

4.1

Learning in the N = S = 2 caseLet us consider the simple example of two agents and two parking spots with pe =pl = 1. As discussed above, player i’s choice of strategy in period t depends on her
attraction for strategy k, Aik , asshown in Eq. (15). Let
A(t) =$A11 (t) A12 (t)A21 (t) A22 (t)%be the matrix of attractions for two agents in period t. Since the attractions evolves
as in Eq. (16), we haveA(t + 1) = ωA(t) + (1 − ω)R(t)where R(t) is the matrix of stimulus agents receive for two strategies in period t,i.e.,R(t) =

$ *

+

1+L 2
k (t) − L [2 − k 1 (t)] + 12 [k 2 (t) − 1][k 1 (t) − 1]
2
* 1+L
+
1
k
(t)
−
L
[2 − k 2 (t)] + 12 [k 1 (t) − 1][k 2 (t) − 1]2

1 2
[k (t) − 1][k 1 (t) − 1]2
1 1
[k (t) − 1][k 2 (t) − 1]2

where the entry in row i and column k corresponds to the stimulus that agent i
receives for strategy k i = k, given that the other player plays k −i (t) ∈ {1, 2}, where−i denotes agent 2 if i = 1 and 1 if i = 2. The probability that agent i choses

15%strategy k = 1 can be written asiProb{ki= 1|t} = P1i (t) =eλA1 (t)1=ii
1 + e−qi (t)eλA1 (t) + eλA2 (t)

(18)

where
q i (t) = λ[Ai1 (t) − Ai2 (t)].
This means that choice behavior depends only on q i (t). The learning dynamics canbe derived for this quantity, taking the difference of the equations for Ai1 (t) andAi2 (t). With a little algebra, one finds,

1 + L −iq (t) = ωq (t − 1) + (1 − ω)λ(2 − k (t − 1))k (t − 1) − L
2
i

i

i

-

(19)

and similarly for agent 2. Notice that q i (t) is a stochastic variable which depends
on the choice k −i (t − 1) ∈ {1, 2} of the other agent, which in turn is drawn from a
distribution which depends on q −i (t − 1).
In the limit ω + 1 the second term is small compared to the first, which meansthat if q i (t) reaches a fixed point value, the stochastic deviations from this will besmall. In other words, it is reasonable to approximate k −i (t − 1) by its expectedvalue E[k −i (t − 1)] = 2 − Prob{k −i (t − 1) = 1} using Eq. (18). This leaves us witha dynamical system for the two variables (q 1 , q 2 ) which has the form,
1+L
1q (t) = ωq (t − 1) + (1 − ω)λ1−
2 1 + e−q−i (t−1)
1 + e−qi (t−1)
ii1Fixed points of these equations are of the general form q 1 = q + z and q 2 = q − zwhere q and z, after some algebraic manipulations, are solutions of the following
equations:,λ (1 − L)eq /2 + cosh zq=2cosh q + cosh z,
λsinh zz=
2 cosh q + cosh z

(20)
(21)

The second of these equations is always satisfied if one sets z = 0. This corresponds to a symmetric equilibrium where agents behave in the same way (q 1 = q 2 ).
In order to study the stability of the solutions, we can introduce the variable16ζ(t) = [q 1 (t) − q 2 (t)]/2. The same algebra as above, leads toζ(t + 1) = ωζ(t) + (1 − ω)λsinh ζ(t)2 cosh χ(t) + cosh ζ(t)(22)

where χ(t) = [q 1 (t) + q 2 (t)]/2. Now, the linear stability of the equilibrium with
χ = q and ζ = z = 0 can be studied through the linearization of Eq. (22) aroundχ = q and ζ = 0. This yields,
λ1∼
ζ(t + 1) = ω + (1 − ω)ζ(t) + O(ζ 2 ).
2 1 + cosh q(23)

Therefore the solution z = 0 is stable as long as
λ1≤ 1.
2 1 + cosh q(24)

The region of stability of the symmetric state, in the plane (λ, L), can be obtainedin parametric form (i.e. varying q) combining Eqs. (20) with z = 0 and (24). The
result is shown in Fig. 1. The increase of λ always drives the system from thesymmetric to the asymmetric state. The behavior of the system has however, a
non-trivial dependence on L: For a fixed λ > 4, the symmetric state is stable either
for small L or for large L. For small L, the symmetric state where both players
choose strategy 1 is stable, while for large L, the state in which both players choose
strategy 2 is stable.

5

Simulation results

The cases with larger Ns are not analytically tractable so that we employ numericalsimulations to analyze the learning dynamics. The model presented in the paper
has eight parameters: the number of agent N , the number of parking spots S, theprobability of each agent coming into the city if she is outside the city, pe , andleaving the city if she is parked in the city, pl . The parameter α determines theshape of the payoff function, L gives the size of loss in case of failing to park before
the city center, λ governs the sensitivity of parking decisions to attractions to parkand not to park. ω represents the importance of past experiences in the evolutionof attraction.
Initially, we randomly assign agents to parking spots, and let each agent leave thespot with probability pl . To ensure that the number of agents who look for a parking
17

1000

Symmetric
100

L

Asymmetric
10

1

0.1
1

10

100

1000

!
Figure 1: Region of stability of the symmetric and asymmetric solutions of thelearning dynamics for the N = 2 case.

18

A. The average strategies
freq.
0.25

B. The average spots

C. The average payoffs
freq.

freq.
0.25

1

0.8

0.2

0.2

0.15

0.15

0.6

0.1

0.1

0.4

0.05

0.05
5

10 15 20 25 30 35 40 45 50

!k"

0.2

5

10 15 20 25 30 35 40 45 50

!s"

"Π$
0.0! 0.1! 0.2! 0.3! 0.4! 0.5! 0.6! 0.7! 0.8! 0.9!
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 2: The distribution of (A) the average strategies, (B) the average spots
parked, and (C) the average payoffs for each agent. Note that s = 1 is the city
center and the larger the s is, the further the spot is from the center. N = S = 50,
pl = 1.0, L = 0.5, α = 1.0, λ = 100.0, ω = 0.9. The mean (the standard deviation)of the distributions of average strategies, average spots, and average payoffs are 22.2
(12.7), 21.4 (12.8), and 0.4 (0.2), respectively.

spot in one period and the number of available parking spots at the beginning ofpl S
the period is the same on average9 , we set pe = N −(1−p. Thus, effectively, we havel )S
one less parameter for the model. In addition, we first fix the number of agents N
to the same as the number of parking spot S such that N = S = 50.10

5.1

Asymmetric outcomes

Figure 2 shows three distributions: (A) the distribution of the average strategies
chosen, (B) the distribution of the average spots each agent parked11 (average parkedspot), and (C) the distribution of the average payoffs each agent received per search12
from a single simulation run.
The parameters are set so that all the parked agents leave the city at the end of
the period, pl = 1.0, the loss of not finding a parking spot before the city center L
is 0.5, the payoff is linear with respect to the relative distance of the spot from thecenter, α = 1.0, players have quite a long memory of past experiences ω = 0.9, andare sensitive to the attractions in choosing their strategies, λ = 100.0.
An estimate of the number of available parking spots is obtained assuming that all parking
spots were occupied in the previous period and that each of them was made available with probability pl , before the present period. This yields Spl free parking spots, and (1 − pl )S agents inthe city parked. Thus, the number of agents outside the city at the beginning of the period is
N − (1 − pl )S and therefore, pe (N − (1 − pl )S) agents will enter the city.
10
Note that when N = S, we have pe = 1.
11
Let ni (s) be the number of times agent i has parked at spot s in the periods under consideration,iand let f i (s) = Pn n(s)i (j) be the relative frequency at which agent i parked at s. Then the averagej(spot agent i parked is < si >= s sf i (s).
12Although the average parked spots do not take into the account agents failing to park before
the center, such cases are included in calculating the average payoff per search.
919

The data are taken from 500 periods, after letting the simulation run for 5000
periods, i.e., from 5001 ≤ t ≤ 5500. Given the parameter values, agents search fora parking spot 500 times during these 500 periods.13
The panel (A) and (B) of Fig. 2 show substantial variations in the strategies andspots chosen by agents. The standard deviations of the distributions the average
strategies chosen and average spots parked are 12.8 and 12.7, respectively. For acomparison, one should note that the size of standard deviation under the maximum
heterogeneity is 14.58 for both distributions.14 One can see from the figures thatwhile there are agents who have chosen strategies such that they park at empty spots
far away from the center (large k), there are agents who have chosen strategies topark very close to the center (small k). Some learned to obtain luckier outcomeswhile others did not. Correspondingly, their average payoffs vary.
While the three distributions shown in Fig. 2 demonstrate the heterogeneity
in the average behaviors of agents, we would also like to know how each agent has
behaved. For example, where has an agent who parked on average at spot 25 parked?
Has she parked in a wide range of spots with similar frequencies or in a few spots
around 25 with very high frequencies and not elsewhere? To see this, we have plotted
the distribution of strategies chosen (panel A) and the distribution of parked spots(panel B) for three agents who obtained the lowest (solid black), median (dashedblack), and the highest (solid gray) average payoff per search in Figure 3. The figure
also shows, in panel C, the distribution of spots that were available when these three
agents searched. Also presented in the figure (panel D) are the basic statistics, suchas the average spot parked and the average strategies chosen, for three agents.
The average payoff per search for the agents with the lowest, median, and thehighest payoffs were 0.10, 0.48, and 0.63, respectively. The figure shows that agents
do not park uniformly across the spots. The distributions of the strategies chosen
by three agents differ substantially from each other. The agent with the lowest
average payoff has chosen strategies leading her to park in the first empty spot and,
therefore, far away from the center (the average strategy was 45). The agent withthe highest payoff, on the other hand, has chosen strategies 4.0, i.e., to park very
close to center, although not right at the center. The agent who received the median
payoff has chosen to park in the first empty spot beyond the half way to the center(average strategy was 20.0.).
As noted above, the highest average payoff per search was 0.63 for this particular
The simulation results show that, given the parameter values, the system reaches stationary
states after 5000 periods. See the Figure 11 in the Appendix.
14
The maximum heterogeneity can be achieved when both distributions are uniform.
13

20

A. Strategies Chosen
freq.
1B. Spots Chosen

C. Spots Available
freq.
1

freq.
1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2
10

20

30

40

k
50

0.2
10

20

30

40

s
50

10

(D) Basic Statistics for Three Agents
Ave. Payoff Ave. Spots Ave. Strategy Success Rates
Highest
0.63
3.1
4.0
0.78
Lowest
0.10
45.0
45.0
1.00
Median
0.48
18.4
20.0
0.86

20

30

40

50

No. of Search
500
500
500

Figure 3: The distribution of (A) strategies chosen by agents, (B) spots chosen byagents, and (C) available spots at the time of search for agents with the lowest(solid black), median (dashed black), and the highest (solid gray) average payoff per
search. The basic statistics for these three agents are shown in (D). N = S = 50,
pl = 1.0, L = 0.5, α = 1.0, λ = 100.0, ω = 0.9. For 5001 ≤ t ≤ 5500
simulation. But this seems to be too low considering that the agent with the highest
average payoff parked on average around spot 3, which should generate an average
payoff close to 0.94. Why is her payoff so low? The reason is because such agents
sometimes fail to park before reaching the center, and obtain the loss −L. The
success rate, the probability of successfully parking before the center, for the agent
with the highest payoff was 0.78. While those for the agent with lowest and median
payoff were 1.0 and 0.86, respectively. These numbers show that agents who parked
closer to the center often succeeded in doing so, but sometimes failed. A quick
calculation verifies the seemingly low payoff. The agent with the highest payoff
obtained, on average, payoff about 0.94 at success rate 0.78, but failed in about 0.22of the times and lost 0.5, yielding an average payoff of about 0.63. The remarkable
point is that, despite of such costly failures, the agent with the highest payoff held on
to strategies inducing parking close to, and did not use the ones leading to parking
far away from, the center.
The large differences in the strategies and spots chosen by these agents, as wellas their payoffs, cannot be explained by the differences in available opportunities.
The panel C of the Figure 3 shows that, at the time of search, there is no major differences, at least on average, in the available parking spots. Thus, one can conclude
that agents learned to behave in different ways and that this resulted in substantial
variations in the outcomes, although they were facing the similar opportunities.
21

s

t=0

t=1

Pk

t=2

Pk

1

Pk

1

1

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.8

0.2

10

20

30

40

k
50

0.2

10

t=3

20

30

40

k
50

10

t=4

Pk

0.8

0.6

0.6

0.6

0.4

0.4

0.4

20

30

40

10

20

30

40

k
50

10

t=7

1

1

0.6

0.6

0.4

0.4

0.2

30

40

50

k

k

40

50

k

1

0.6

20

50

0.8

0.4

10

30

Pk

0.8

0.2

20

t=8

Pk

0.8

40

0.2

t=6
Pk

k

0.8

0.2

10

50

1

0.8

k
50

40

Pk

1

0.2

30

t=5

Pk

1

20

0.2

10

20

30

40

50

k

10

20

30

Figure 4: Initial dynamics of probabilities of choosing each strategy for 3 players
shown in Fig 3. The colors correspond to that in Fig 3 as well. Note that for t = 0,
all the players are identical.

22

To better understand the dynamics of individual behavior, the initial dynamics of the probabilities with which these three agents choose their strategies, Pki (t),are shown in Figure 4. As noted above, all these agents are identical at the very
beginning of the simulation, t = 0. Thus, they all have the identical probability distribution over strategies. As shown in the figure, agents first choose their strategies
randomly. Based on the obtained payoffs and chosen strategies, they change theirstrategy choices. For example, the agent with the highest average payoff during
5001 ≤ t ≤ 5500 (shown in gray) has chosen k = 4 in period 0 and obtained a highpayoff. As a result, he learned to play k = 4 with a probability close to one fromt = 1 on. But the probability of this agent choosing strategy 4 becomes a bit lower
in t = 4. This is because the agent failed to find a spot in period 2 and 3. But theagent tries with strategy 4 again in period 4 and was successful in parking at spot
3. As a result, the probability of her choosing strategy 4 rose again. Although she
failed to park once more in the following period, her probability of choosing strategy
4 remained high. What we can observe here is that once an agent learns to play astrategy with small k, unless she failed to park before the center too frequently (orthe loss of doing this is too large), she continues to play such strategy.
The results for the agent with the lowest average payoff during 5001 ≤ t ≤ 5500
(shown in solid black) is more complex. The agent has first chosen strategy 46 andparked in the spot 46. As a result, her probability of choosing the same strategy
became a little bit higher than the other strategies (see t = 1). Next, she has chosen
strategy 41 and again successfully parked at spot 41 (t=2). In the next period,
she has chosen strategy 33 and parked at spot 8. This results in putting higher
probabilities on choosing strategies k ∈ [8, 33] (t=3). Among which, she chose
strategy 14 in the following period and parked at spot 14 resulting in placing a very
high weight in choosing strategy 14 (t=4). In the following period, he chose strategy
14 but failed to park. As a result, the relative weights on strategies k ∈ [1, 14] become
lower while those for k ∈ [15, 33] become higher (t=5). He then chose strategy 27
and parked at spot 27 (t=6). The agent continues to choose strategy 27 and parkat the spot 27 for a while. But as one can see from the result shown in Figure 3,
she learned to park even further away in later periods.

5.2

Dependence on parameters

The previous subsection has shown, for a particular set of parameter values, thatagents who are initially identical, learn to behave differently and sort themselves
into parking at the different spots along the street. Some agents learned to park
23

Heterogeneity of Parked Spots

Heterogeneity of Strategies

Σ#s$

Σ#k$
14

12

12

10

10

8

8

6

6

4

4

2

2
1

10

100

Λ

1

10

100

Λ

Figure 5: The effect of λ on the heterogeneity of parking behaviors. Dashed: L = 0.5,
Black: L = 1.5, Gray: L = 2.5. The heterogeneity of behavior is measured by
the standard deviation of the distribution of the average parked spots (left) andby the standard deviation of the distribution of the average strategies (right) . The
statistics are based on taking average from 100 simulation runs. Error bars represent
one standard deviation bound from the 100 simulations. N = S = 50, pl = 1.0,
α = 1.0, ω = 0.9.

far away from, while the others learned to park near to, the center. That is, somelearned to be unlucky while others learned to be lucky. How do such results dependon the parameters of the model? This subsection analyzes the dependence of theresults on the parameter values.
The simple 2 players case we have considered above shows that there is a criticalvalue of λ beyond which we will see asymmetric outcomes. The analysis also showsthat the result depends on L. Therefore, we first consider the dependence of theresult on λ and L. As one can easily imagine, when λ = 0, agents always choose
strategies randomly regardless of the values of propensities, and thus, we will notobserve any heterogeneity in the behavior of agents. What happens for larger valuesof λ? Figure 5 shows the standard deviation of the distribution of the average parked
spots, σ
 (left) and the standard deviation of the distribution of the average
strategies, σ<k> (right) for various values of λ. We consider three values of the loss
from failing to park before the center, L = 0.5 (dashed black), L = 1.5 (solid black),and L = 2.5 (solid gray).15 Each point in the figure is the average value from 100
simulation runs.
The figure shows that heterogeneity increases very sharply with λ for 10.0 ≤λ ≤ 100, and then it gradually reaches a plateau. This pattern can be observed
for all three values of L. The figure also suggests that L may have an effect on
the degree of heterogeneity for λ ∈ [20, 100] but less when λ is large enough. To
Other parameters are fixed as in the previous section (Namely, N = S = 50, pl = 1.0, α = 1.0,
ω = 0.9). The data are taken from the 500 periods after the first 5000 periods has past, i.e., from5001 ≤ t ≤ 5500.
15

24

Heterogeneity of Parked Spots

Heterogeneity of Strategies

Σ"s#

Σ"k#

14

14

12

12

10

10

8

8

6

6

4

4
2

2
0.1

1.0

10.0

L

0.1

1.0

10.0

L

Figure 6: Effect of L on heterogeneity of parking behavior. Dashed Black: λ = 50.0,
Dashed Gray: λ = 100.0, Black: λ = 150.0 Gray: λ = 200.0

see this differently, Figure 6 plots heterogeneity of parked spots and heterogeneity
against various values of the loss, L, for four values of λ: λ = 50.0 (dashed black),
λ = 100.0 (dashed gray), λ = 150.0 (solid black) and λ = 200.0 (solid gray). As in
the previous figure, each point in the figure is the average value from 100 simulation
runs.
The figure shows that for λ = 150 and λ = 200 the results are not distinguishable
for all values of L ∈ [0.0, 2.5]. The basic pattern is that heterogeneity first increases
slightly with L, and in the interval 0.5 ≤ L ≤ 2.0, it either decreases gradually or,in some cases, stays constant (heterogeneity of strategies for λ ∈ {150, 200}). For
larger values of L, the degree of heterogeneity declines quite drastically for all valuesof λ considered here. The lower degree of heterogeneity under large L is in line withthe analytical results for N = S = 2 case shown above. The fact that heterogeneity
is high even at L = 0 is in line with our equilibrium considerations for large N .16
As noted in the previous section, when L is very high, agents learn to choose higher
k strategies, and the rate at which agents fail to park before the center is lower.
These figures show that as long as λ large enough, and L not too high, we havevery high degree of heterogeneity in the behavior as a result of learning.17 Thisshould be contrasted with the equilibrium analysis in previous section. There, theresult showed that the larger the value of L is, the easier it is to sustain heterogeneous
outcome.
Recall that the equilibrium analysis for pl = pe = 1 suggests that for N > 2, even with L = 0
we should expect a high degree of heterogeneity.
17
One should note that these results are from 500 periods. Because of all the strategies always
have non-zero probabilities of being chosen, in a very long run, strategies chosen by agents maychange. As shown in Figure 12 in the Appendix, it is indeed the case. If we consider much longer
periods (say 25000 periods) over which to observe the behavior of agents and measure heterogeneity,then degree of heterogeneity will be smaller, although it is still much larger than for the λ = 0.0
case.
16

25

L = 0.5

L = 1.5

0.9
0.7
0.5

L = 2.5

0.9
0.7
0.5

Ω

0.9
0.7
0.5

Ω

0.3

Ω

0.3

0.1

0.3

0.1

10

0.1

10

10

Σ$k%

Σ$k%
5

Σ$k%
5

5

0

0

0.25

0

0.25
0.5

0.25
0.5

1.0

Α

0.5
1.0

2.0

Α
4.0

1.0
2.0

Α
4.0

2.0
4.0

Figure 7: Standard deviations of the distribution of average strategies (σ<k> ) for
various combinations of α ∈ {0.25, 0.5, 1.0, 2.0, 4.0} and ω ∈ {0.1, 0.3, 0.5, 0.7, 0.9}
for three values of L: L = 0.5 (left), L = 1.5 (middle), L = 2.5 (right). σ<k> isthe average from 100 simulation runs. For all the plots λ = 100.0, N = S = 50,
pl = 1.0, and data are taken from 5001 ≤ t ≤ 5500.
How do the results depend on other parameters? Let us now consider ω and
α. Figure 7 show the heterogeneity of parking behavior (measured by the standard
deviation of the distribution of the average strategies chosen, σ<k> ) for various
combination of α and ω. Three values of L, L ∈ {0.5, 1.5, 2.5} are considered. Whatfigure shows is that ω has to be large for there to be a high degree of heterogeneity
in the behavior. The critical ω for which we obtain a high degree of heterogeneity
increases with L.
The figure also shows that heterogeneity emerges more easily when α is high.
This is reasonable since high α means that most of the spots are as good as parking
near the center. On the contrary, when α is small, then most of the spots are as bad
as parking at the furthest spots, thus, we see much lower degree of heterogeneity.
One of the reason for running extensive sets of simulations was to see how the
probability at which those agents who are in the city leave, pl , affects the resulting
outcome. This was because the equilibrium analysis produced quite different resultsin the two extreme cases, namely the one where pl and pe are very small and theother where pe = pl = 1. In particular, the equilibrium analysis suggested thatwhen pl = pe = 1 we should not expect to see homogeneous outcomes except forvery special values of parameters, while when pe and pl are very small, homogeneous
outcomes always exists.
To see the effect of pl , Figure 8 shows the heterogeneity of parked spots, σs (left),and chosen strategies σk (right), for various pl and three values of L ∈ {0.5, 1.5, 2.5}26

Heterogeneity of Parked Spots

Heterogeneity of Strategies

Σ"s#

Σ"k#

14

14

12

12

10

10

8

8

6

6

4

4
2

2
0.05

0.1

0.5

1.0

pl

0.05

0.1

0.5

1.0

pl

Figure 8: Effect of pl on heterogeneity of parking behavior. Dashed Black: L = 0.5,
Dashed Gray: L = 1.5, Black: L = 2.5. For all the simulation, N = S = 50,
λ = 100.0, α = 1.0, ω = 0.9(dashed black, dashed gray, and solid black). Other parameters are set so that
λ = 100, α = 1.0, and ω = 0.9. It can be observed easily from the figure, there isan increasing relationship between pl and the resultant heterogeneity. Namely, thehigher the pl is, the greater the σk (and thus, σs ) becomes.18

5.3

Heterogeneous AgentsIn the previous section, we considered cases where agents are ex ante identical,
and showed that heterogeneity in the behavior emerges as agents learn. We havealso shown that the emergence of heterogeneous behaviors depended, partially, onparameter values. For example, when λ is too low, we did not see differences in the
behavior of agents. What happens if agents are not identical? In particular, what
happens if there are agents with a low λ and a high λ? Since low λ agents behave
quite randomly, such random behavior may interfere with how agents with the highλ learn to behave. We have experimented with half of the agents having λ = 0.0,
who thus behave randomly, and the other half having λ = 100.0 for three values of
α holding other parameter values the same as in Figure 2 and 3.
The results are shown in Figure 9 and 10. Figure 9 shows the distribution of
the average strategies chosen and the average spots agents parked. As before these
two looks quite similar. The spike around < k >= 25 and < s >= 24 is due tothe existence of agents with λ = 0.0. Since these agents choose strategies randomly,
on average, their chosen strategy will be close to 25. A part from the peak, theresulting distributions are still flat, suggesting there are lucky and unlucky agents.
Note that when pl < 1, agents do not search for parking every period. In order to make thenumber of searches, at least on average, as similar as possible across various pl , we have taken datafrom 5001 ∗ p−1≤ t ≤ 5500 ∗ p−1
l
l .
18

27

A. The average strategies

B. The average spots

freq.
0.25

freq.
0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05
5

10 15 20 25 30 35 40 45 50!k"5

10 15 20 25 30 35 40 45 50!s"Figure 9: The distribution of (A) the average strategies chosen and (B) the average
spots parked.Half of the agents have λ = 0.0 and the other half have λ = 100.0.
N = S = 50, pl = 1.0, L = 0.5, α = 1.0, ω = 0.9.

In particular, as Figure 10 shows, the agent with the highest (lowest) average
payoff always parks close to (far away from) the center as in the case where all theagents had λ = 100.0. These two agents had λ = 100.0. One the other hand, theagent with the median payoff had λ = 0.0, and parked almost uniformly randomly
across all the parking spots. The figure shows that the heterogeneity in the behaviors
emerges even in the presence of randomly behaving agents.

6

Summary and Conclusion

In this paper we consider the idea that luck may not be totally exogenous. In a number of situations, otherwise identical individuals can find themselves in very different
states. Those who are in the worse positions can be considered as “unlucky”, yet if
the choices available are unequal, then any allocation must treat people asymmetrically. It could well be the case that all people choose the same strategy and someby pure luck get the better opportunities and others do worse. What interested us,
however, was whether it could be the case that some people choose, ex ante, to bein less favorable positions. Thus it is not the luck of the draw but the conscious
choice of the individuals to be well or poorly treated.
Although we have chosen a simple parking problem as our basic example, there
are many other cases in which individuals have to choose amongst a set of alternatives, whilst other are doing the same and there is no possibilty of going back to theones that have been rejected. One has only to think of dating problems, to see thatgoing back to the best alternative amongst those one has gone out with might beproblematical. In a more serious vein it is not always possible to retake a job offerthat one has previously refused and it may well be the case that other jobs have28

A. Strategies Chosen

B. Spots Chosen

freq.
1.0

freq.
1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2
10

20

30

40

50

k

10

20

30

40

(c) Basic Statistics for Three Agents
Ave. Payoff Ave. Spots Ave. Strategy Success Rates
Gray
0.72
1.85
2.0
0.82
Black
0.15
42.4
43.0
0.99
Dashed
0.43
24.2
26.5
0.91

50

s

No. of Search
500
500
500

Figure 10: The distribution of (A) the strategies chosen and (B) the spots parked foragents with the lowest (solid black), median (dashed black), and the highest (solidgray) average payoff per search. Half of the agents have λ = 0.0 and the other half
have λ = 100.0. Note that s = 1 is the city center and the larger the s is, the further
the spot is from the center. N = S = 50, pl = 1.0, L = 0.5, α = 1.0, ω = 0.9.

already been taken.
Thus while our example is, of course, special it can be re-interpreted to cover anumber of other economic situations. In our case, for some individuals to choose
inferior options leaving the better ones for others can be an equilibrium of theassociated game. Whilst this is interesting in the one-shot case, it is importantto understand what happens in a repeated situation. The question we posed waswhether, in such a case, the individuals can learn to be “unlucky” or “lucky”. Thatis, we were interested in whether some individuals learn to systematically make thepoorer choices, and thus always leaving the better opportunities open and letting
others learn to be “lucky”.
We showed analytically, in simple cases, that this can happen. Both symmetricand asymmetric equilibria can arise, depending on features of the environment where
agents interact. One key parameter is the frequency (pe , pl ) with which agents seek
and leave parking spots, i.e. parking turnover. We identify two polar situationsin the static analysis of Nash equilibria. When parking turnover is slow (pe , pl &1), we find that asymmetric outcomes, when N is large, happen either when thecost L is extraordinarily large, or when the number of free spots is small, in thestationary state. At the opposite extreme, we also examine the fast turnover case,when everyone searches for parking spots every period (pe = pl = 1). In thatcase, we find that symmetric Nash equilibria exist only when α and L are very29

small. At the same time, we have also shown that unless the cost of not parking
before the center, L, is large enough, the completely asymmetric outcome is not
a Nash equilibrium. This implies that in the fast turnover case, some degree ofheterogeneity is expected under generic conditions, and it identifies fast turnover as
one of the conditions favoring the emergence of “luck” as a discriminating factor
across agents. Interestingly, simulations show that this intuition also holds in more
complicated cases.
A second interesting point of our analysis concerns the incidence of the cost
L on the resulting outcome. The analysis of learning in the case of two agentsshows that asymmetric outcomes only occur in an intermediate range of values of L.
When L is low (high), the symmetric outcome where both agents choose strategy 1(2) is stable. Thus, when L is low, the street parking spots are not fully utilized,
while when L is higher, they are fully utilized either because the two players choose
different strategies, or because both players choose the strategy to park further away.
This result is also confirmed by numerical simulations for a larger number of agents.
When the number of agent is large, we do not have symmetric outcomes for low L,
instead we have high degree of heterogeneity even with L = 0. However, for large
enough values of L, the degree of heterogeneity becomes low, and agents learn tochoose strategies with higher k, i.e., to park further away from the center. That is,when L is very high, it is less likely that we observe “lucky” agents. This is in quite
sharp contrast to the equilibrium result that shows the higher the L is, the more
likely it is to have asymmetric outcomes. However, both in the equilibrium analysis
and in the learning dynamics, high enough values of L lead to the efficient use ofstreet parking.
One last remark is in order. An alternative explanation for the sort of phenomenathat we have analysed is that people simply have different intrinsic aversions to risk.
However our arguments suggest that these attitudes to risk may well be learned
rather than inherent. This is why we claim that the idea that some people are“born under a lucky star” does not always stand up to scrutiny.

30

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32

A

The asymmetric equilibrium for pe, pl & 1

In order to derive the stability condition (4), let us consider the case ki = i. The
agent with the largest incentive to deviate is clearly the worse off, i = N . Let
V (k) = E[uN (s)|kN = k] be the expected payoff of agent N if she deviates to purestrategy k. Then
V (k) = ek π(k) + (1 − ek )V (k − 1),

ek = P {k is free}(25)

because if parking s = k is occupied, agent N ’s situation is the same as if she hadchosen strategy k − 1. Clearly V (N ) = π(N ) = 0 and V (0) = −L, because if there
is no empty parking, agent N will have to pay cost L. In the Nash equilibrium weare studying, each position is occupied by a specific agent, so the probability ek = hthat spot k is free is the probability that the corresponding agent is not in the city.
The solution of Eq. (25) is easily obtained introducing generating functions for V (k)and π(k) and it reads
V (k) = hk#j=1

(1 − h)k−j π(j) − (1 − h)k L(26)

It is clear that this is largest for k = N − 1. The condition for the stability of the
Nash equilibrium can be recast as a condition on the cost agent N incurs if there isno free space, which is Eq. (4).

B

The symmetric equilibrium for pe, pl & 1

Let xs = 1 if spot s is empty and xs = 0 if it is empty. In the stationary state, thedetailed balance condition reads
P {xs = 1}W {xs = 1 → 0} = P {xs = 0}W {xs = 0 → 1}
where W are the transition rates (probability per unit time). Now, es = P {xs =0} = 1 − P {xs = 1} and W {xs = 1 → 0} = 1, which is just the rate at which
agents leave the city. The rate W {xs = 0 → 1} requires more work. The number
W {xs = 0 → 1}dt of agents who reach site s in a time interval dt, is proportional
to the number N h of agents not in the city, times the probability ηdt that each ofthem enters the city in time interval dt, times the probability qs that she actuallyreaches spot s (see below). In other words, W {xs = 0 → 1} = N ηhqs . Combining
33

these relations we havees =1
1 + N (1 − h)qs

(27)

In order to compute the probability qs that an agent arrives at spot s we have toconsider all those events where she aims at some k ≥ s and finds no empty place
before s. If σk is the probability of choosing to start at site k, we haveqs = σs + (1 − es+1 )σs+1 + (1 − es+1 )(1 − es+2 )σs+2 + . . . + (1 − es+1 ) · · · (1 − eN )σN= σs + (1 − es+1 )qs+1(28)

This allows us to derive a backward equation for es in terms of σk :-−1,(1 − es+1 )2
es = 1 + N (1 − h)σs +
es+1

(29)

On the other hand, agents maximize their expected utility
V (σ) =

N#
s=1

with the constraint(

qs (σ)es u(s) − Lq0 (σ)

k σk = 1. Taking into account thatk.
∂qs
= θ+ (k − s)
(1 − ej )
∂σk
j=s+1

where θ+ (k) = 0 if k < 0 and θ+ (k) = 1 for k ≥ 0, the marginal utility of choosing
to start at k is
k
k
k
#
.
.
∂V
(1 − ej ) − L (1 − ej ),µk ==u(s)es
∂σk
s=1
j=s+1
j=1

k = 1, . . . , N.

(30)

Notice that:µk+1 − µk = ek+1 [u(k + 1) − µk ](31)

The optimality condition implies that σk > 0 on the set of values of k whereµk is maximal. Let us analyze Eq. (31) in detail. At k = 1, we have µ1 =u(1)e1 − L(1 − e1 ) = u(1) − (1 − e1 )[L + u(1)]. So the curve µk vs k starts below
the curve u(k) for small k. As long as this is true, µk increases with k, as indicatedby Eq. (31), whereas u(k) decreases with k by hypothesis. There should be a pointk ∗ where the two curves cross (i.e. u(k ∗ ) > µk∗ −1 and u(k ∗ + 1) < µk∗ ) and this is34where µk attains its maximum. Generically, for a given es , the maximum will beattained on a single point k ∗ . Therefore, the best response will be σk = δk,k∗ . The
only problem which remains is that of characterizing the threshold k ∗ as a functionof η, u(k) and L. Much progress can be made by taking the limit N → ∞. First
we set1,(z),N= m(z),es =z = s/N(32)µk

z = k/N(33)

u(k) = ν(z)(34)

k∗ = N z ∗(35)Then Eqs. (29,31) can be cast in the formd,= −,2 ,
0 ≤ z ≤ z∗dzdm
= ,(ν − m)
dz

(36)
(37)

The first is easily integrated with the initial condition ,(z ∗ ) = 1/(1 − h) = 1 + 1/η
and it yields,(z) =1+η
1=,∗
η − (1 + η)(z − z)z0 + z

0 ≤ z ≤ z∗.

(38)

with z0 = η/(1 + η) − z ∗ The second is also easily integrated and it gives (note thatm(0) = −L)
/ z

Rz

Rz

due− u dy#(y) ν(u),(u) + m(0)e− 0 dy#(y)
0
,/ z
1
#
#
=
dz ν(z ) − Lz0z0 + z 0m(z) =(39)(40)

the condition for z ∗ is now ṁ(z ∗ ) = 0 or u(z ∗ ) = m(z ∗ ). This gives/ z∗

dz # [ν(z # ) + L] =0

It is easy to show thatz∗ ≤η[ν(z ∗ ) + L]
1+η

η
=1−h
1+η

(41)

(42)

is less then the fraction of agents at work, and also the number of empty slots left

35

Standard deviation of the distribution of
the average spots
the average strategies
Σ"s#

Σ"k#

14

14

12

12

10

10

8

8

6

6

4

4
2

2
50000

100000150000t

200000250000

50000

100000150000200000t

250000Figure 11: The time series of the measures of heterogeneity.

for s ≤ k ∗ is

∗

k
#
s=1

es =

/ z∗
0

,

η
dz,(z) = log
η − (1 + η)z ∗

-

(43)

This means that a fraction 1 − h − z ∗ of the agents does not find an empty spotand has to pay the cost L. Therefore the expected utility of an agent in this Nashequilibrium is/z∗

Es [u] =0dzν(z) − (1 − h − z ∗ )L(44)

where the first term is the utility of parked agents and the second is the cost of those
who didn’t find a place19
The same calculation for the asymmetric Nash equilibrium, when it is stable,
gives
/
1

Ea [u] = (1 − h)dzν(z)(45)

0

so for fixed L and large N , we know the asymmetric Nash equilibrium is stable onlyin the limit h → 0 and in this limit it provides higher utility to agents.

C

Stationarity of simulation result

The results shown in main text are all based on the statistics taken from 5001 ≤t ≤ 5500 (for pl = 1.0 and adjusted appropriately for other values of pl ). One
may question whether the results we are reporting is transient or not. As shownby Figure 11, our measure of heterogeneity is quite stable, and therefore, one canconsider that initial 5000 periods were enough for the simulation to reach “steadyTo get this, pick an agent at random: with probability h, she’s not in the city, and thecontribution to the utility is zero. With probability (1 − h)(1 − es ) + 1 − h she is parked in spots ≤ k ∗ and otherwise she is paying the cost L.
19

36

Standard deviation of the distribution of
the average spots
the average strategies
Σ"s#

Σ"k#

14

14

12

12

10

10

8

8

6

6

4

4
2

2
50000

100000150000200000

NumberofPeriods

50000

100000150000200000

NumberofPeriods

Figure 12: Long run outcomes. The standard deviation of the distribution of theaverage parked spots over many trials. λ = 100, pl = 1.0, L = 0.5, α = 1.0, ω = 0.9.

state.”

D

Over longer periods

Figure 5 shows that for λ above certain value, we obtain heterogenous behaviors
among agents. So far, the data were taken from the 500 periods after 5000 periods have past from the beginning of the simulation. What happens if we consider
outcomes from longer intervals? Because of the randomness in the strategy choices,
we expect that the longer the time horizon we consider, the less heterogeneous the
behaviors of the agents become.
Figure 12 shows the σ
 and σ<k> for various number of periods20 up to 250000
periods for λ = 100. One can see that the longer the periods we consider, the lower
the degree of heterogeneity becomes. But as noted, this is unavoidable because of
the randomness in strategy choices.

20

All after the initial 5000 periods have past.

37