Higher order beliefs in the financial market : equilibrium of winning number and performance analysis in a repeated p-beauty contest game

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Higher order beliefs in the financial market

Equilibrium of winning number and performance analysis in a

repeated p-beauty contest game

Andrea Rusman

Student number: 5887313

Date of final version: August 16, 2015 Master’s programme: Econometrics

Specialisation: Game Theory Supervisor: Dr. M. van Rooij Second reader: Dr. M. Koster

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Abstract

In financial markets people have to form beliefs about what other people think about the price of an asset and what other people think about what yet other people think. These forms of beliefs are called higher order beliefs. The purpose of this study is to analyse the e↵ects of newcomers in financial markets where higher order beliefs are formed. To this aim, a repeated p-beauty contest game (p-BCG), a game in which players have to form higher order beliefs, is modelled. In this model players can learn from previous rounds and new inexperienced players are introduced after each round. The path of the winning number of this game and the performance of the players are analysed in order to see what happens to the price of an asset and the performance of traders, in relation to di↵erent learning strategies and di↵erent ways of forming higher order beliefs.

From a theoretical point of view it can be said that the Nash equilibrium in the p-BCG is zero. However, I have shown that this is not an evolutionary stable strategy, which implies that entrants can destabilize this equilibrium and newcomers in financial markets can steer an asset price away from its fundamental value.

In this thesis the behaviour of a group of people, playing the p-BCG was modelled. From my model six things can be concluded, about the deviation from the Nash equilibrium, the variance of the winning number, the way in which players form higher order beliefs, and the performance of the players for di↵erent information sets. All of these conclusions can be translated to financial markets.

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I would like to express my gratitude towards my supervisors Marieke van Rooij and Maurice Koster, for investing a lot of time in my project and always helping me forward when I got stuck. Furthermore I would like to thank my friends for engaging in conversations with me about my thesis, forcing me to think critically about the questions at hand. Lastly, I want to say a special thanks to John Forbes Nash Jr. for sparking my interest in Game Theory and inspiring my switch from mathematics to econometrics.

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Introduction

1.1 Higher order beliefs in financial markets

In financial markets asset prices today depend on the expectations about asset prices tomorrow. For traders, success in these markets may depend on how well they anticipate the beliefs of other traders, how these other traders anticipate the beliefs of yet other traders, etc. In other words, their success depends on the formation of ’higher order beliefs’: A strategy in which a person does not only take into account what other people believe, but also what other people believe about yet other people’s beliefs and so on. In order to respond with the best possible reply, one has to understand how other people form these higher order beliefs. For instance, if other traders only form their beliefs based on their own guess, your best reply is to what you expect the average beliefs to be. If however, traders already form their guess based on the expected average beliefs, your best reply is to the average of what you expect of the expected average beliefs and so on. In sum, traders have to anticipate on the actions of everyone around them. The mainstream financial literature largely ignores the issue of traders forming such higher order beliefs (Allen and Morris, 1998). It is a challenge to investigate how people form these higher order beliefs. Keynes (1936) has already suggested that higher order beliefs are very influential, and introduced a metaphor of the market as a beauty contest game. In this game competitors have to pick the six prettiest faces from a hundred photographs. The competitor who picks the faces that most nearly correspond to the average preferences of all competitors is the winner. In other words, competitors have to anticipate what the average opinion expects the average opinion to be.

Even though Keynes’ idea is nearly 80 years old, the higher order belief model, where people’s actual behaviour is taken into account, is still subordinate to the rational expectations models. These are the models in which agents’ expectations equal the true statistical expected values (Muth, 1961). Therefore, in the rational expectations model the outcome is indisputable ac-cording to the model. In the higher order beliefs model however, some assumptions have to be

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made on the amount of iterations of people’s beliefs. This is in accordance with the problem in the entire field of bounded rationality. If we assume people are rational there is only one right answer (Conlisk, 1996). If we let go of this rationality, suddenly there are several if not infinitely many answers. However, it has been shown that in a model of higher order beliefs, data of the American stock market does indeed fit better than it does in rational expectations models (Monnin, 2004). To conclude, even though a higher order beliefs model will translate reality better, it is also more difficult to use, and thus not yet widely used.

1.2 Experience in the financial markets

When asset prices are formed by higher order beliefs, traders might benefit from gaining ex-perience on the behaviour of other traders. Slonim (2005) has indeed shown that, in a beauty contest, experienced players earn more than inexperienced players. These findings have many implications. For instance, they imply that it might not be beneficial for financial institutions to have a large throughput of employees, because experience and thereby earnings will be lost. A more pressing matter however, is that the loss of experience could lead to more bubbles and crashes in the stock market (Deck et al., 2014). It has also been shown that, when an individual has access to both private and public information about an asset’s payo↵, then the public information is a better predictor of the average opinion (Allen et al., 2006). In a higher order beliefs model investors therefore show an excessive reliance on public information, because it is a better predictor of the price of an asset. This induces a deviation of the mean price path of the asset from the fundamental price, which can lead to bubbles and crashes (Allen et al., 2006). On the other hand, experience usually produces more private information, which drives the asset price towards its fundamental value. Therefore, when many inexperienced traders are active in an asset market, experienced traders will react by mostly using public information about an asset’s payo↵, thereby driving its value away from its fundamental value.

In recent years however, partly due to the financial crisis of 2008, there have been many layo↵s in the financial sector. For example, in the Netherlands, just recently thousands of people got laid o↵ from the large bank ABN Amro (Giebels and Ho↵s, 2015), while it prepared for a stock issue. It is therefore important to get a better idea of the consequences of newcomers.

Aside from this, the financial sector already has a structurally large throughput of employees. For example, every year a few percent of employees (this percentage di↵ers across banks, but is for instance 3% at Goldman Sachs (Luyendijk, 2015)) of the major banks in The City, the financial district of London, are being replaced and few people work in the sector for more than 10 years, due to the stressful work environment and long hours (Luyendijk, 2015). Moreover, according to Luyendijk (2015), an anthropologist who spent two years in The City researching the financial sector, people work on average between 18 months and three years. This is a relatively new development. Former senior official of Citibank says: “In my day [1960’s] if you joined Citibank you were expected to stay at Citibank all your life.” (de Freytas-Tamura, 2012)

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CHAPTER 1. INTRODUCTION 3 Thus even though experience is clearly important in the financial sector, there exists a large throughput of employees. Newcomers can induce a deviation from the fundamental price and an increase in bubbles and crashes. In order to analyse the magnitude of this problem a model of a game will be made, which will be repeated several times, such that newcomers can be introduced. It will be a model of the ‘p-beauty contest game’ (p-BCG), because in this game, as in financial markets, contestants are required to form higher order beliefs.

1.3 P-beauty contest game

As mentioned before the p-BCG is a metaphor for financial markets because players have to form higher order beliefs about the strategies of other players. Because I am interested in the e↵ect of newcomers, the game will be played for many subsequent rounds, such that players can gain experience and some experienced players can be replaced by inexperienced players. In this game every one of the N players must pick a number from the interval [0, 100] and whoever comes closest to p times the average wins the game (where 0 < p < 1) and receives a payo↵. As we shall see in the next chapter, in this game there is a unique Nash equilibrium of zero, which implies that this is the rational strategy. This is the rational strategy if everyone is fully rational and everyone knows that everyone is fully rational. If however, you are rational but you have no information on the rationality of your opponents, choosing zero might not be the rational option.

Indeed, many experimental studies show that most people use the boundedly rational strategy of finitely iterated domination (Bosch-Domenech et al., 2002). For instance, when a player iterates twice, he will choose approximately p2_{· 50. Nagel (1995) was the first to place this game in an} experimental setting and found that players did indeed use forms of higher order beliefs. Thus, instead of choosing the Nash equilibrium players chose to iterate finitely many times. This is in accordance with behaviour that can be observed in financial markets, where traders often-times deviate from the fundamental price and instead use some form of higher order beliefs (Allen et al., 2006).

When the p-beauty game is repeated and players can learn from previous rounds and update their strategy, the winning outcome of the game goes towards the equilibrium outcome of zero (Nagel, 1999). In the financial sector however, experienced players are constantly replaced by inexperienced players, which may result in stopping the winning outcome of the game from going towards the Nash equilibrium. This can result in an asset price moving away from the fundamental price and creating bubbles and crashes. Therefore this thesis will focus on the following research question: What is the path of the winning number and the performance of the players in the p-beauty contest game when players are able to learn from previous rounds and new inexperienced players are introduced after each round?

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In order to answer this question I intend to model the p-beauty game which will be played for many subsequent rounds, in which players can gain experience and some experienced players will be replaced by inexperienced players after each round. The aim is to understand what happens to the path of the winning outcome with di↵erent kinds of information sets available to the players, and specifically whether the winning number will still go towards the Nash equilibrium. Furthermore, the aim is to conclude whether experience has influence on the performance of the players.

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Chapter 2

Theoretical background

2.1 Beauty contest in financial markets

Financial markets can be studied by using the analogy of the beauty contest, in which people have to form higher order beliefs. Many papers have been written on asset prices in a beauty contest setting [Allen et al. (2006), Monnin (2004), Slonim (2005), Allen and Morris (1998), Crawford et al. (2010), Biais and Bossaerts (1998), Kocher et al. (2009)]. For instance, Allen et al. (2006) have shown, by developing a framework that can accommodate higher-order beliefs, that the mean path of prices in a stock market will deviate from the expected fundamental value of the asset (thus the ‘rational’ value), which can induce bubbles and crashes. Moreover, Slonim (2005) shows that when inexperienced players enter the p-BCG, or in his analogy an asset market, players quickly learn to condition their own strategies on the experience level of others, thereby causing the path towards the Nash equilibrium to stop before having reached it. In order to analyse the e↵ect of newcomers in financial markets, I will therefore use the p-beauty contest game.

2.2 The p-beauty contest game

The p-BCG has been studied experimentally in several di↵erent settings. It was first introduced by Moulin (1986) to illustrate how an equilibrium can be obtained by iterating deletion of weakly dominated strategies, a technique in which a strategy is deleted from the action space because it is dominated by at least one other strategy, but then in the new strategy set, some strategy can again be deleted, etc. Nagel (1995) was the first to study the game experimentally and found that most people did not choose the Nash equilibrium, but instead used some level of iterated best response. In other words, people used some form of higher order beliefs.

Later, this conclusion has been supported by an experiment by Bosch-Domenech et al. (2002). They organized a contest of the p-BCG in three newspapers, thereby obtaining the choices

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Figure 2.1: Results of the newspaper challenge as described in Bosch-Domenech et al. (2002). Peaks can be observed for the level-1 (at 33), level-2 (at 22) and rational strategy.

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CHAPTER 2. THEORETICAL BACKGROUND 7 of the thousands who participated. In figure 2.1 (next page) you can see the results of this experiment. The p in this case was 2₃. In all three newspaper contests there are three dominant strategies. First, the best response to the average of 0 and 100 (level-1), being 33. Second, the best response to level-1, thus 22. Last, the Nash equilibrium of zero. From these findings we can conclude that most people use the strategies of higher order beliefs of level 1 or 2, or the ‘rational’ choice of zero (it is only rational if you know that everyone else is also rational).

2.2.1 Strategies

One of the questions of the p-BCG is how people iterate in forming their higher order beliefs, and thereby their strategies. Ho et al. (1998) define a level-k player as a player who performs k iterations. This model implies that a level-k player responds to the assumption that all other players are level-(k-1) players, thus choosing the best response of pk_{· 50. Most players are a} level-1 or level-2 player (Ho et al., 1998), and no one will iterate more than four times (Camerer et al., 2004). Camerer et al. (2004) introduced a di↵erent model for the way in which people form the assumption on the strategies of other players, the so-called cognitive hierarchy (CH) model. In this model, players play their best response to the assumption that all other players are Poisson distributed over all lower level strategies. This is in accordance to psychological evidence of overconfidence about one’s own skillset relative to others’ skillsets (Camerer et al., 1999).

In sum there are two influential models for how people iterate in forming their higher order beliefs. In order to investigate the e↵ect of the di↵erent ways in which people form their beliefs, I will model both the ‘level-k strategy’ and the cognitive hierarchy strategy. Moreover, in my model, I will use three di↵erent strategies based on the information sets available to the player.

2.2.2 Learning possibilities

Apart from the way in which people form their higher beliefs, and subsequently which strategy they use, another question is how players will learn.

In the model in this thesis everyone will learn by using the winning number of the last round, in order to arrive at their new guess. The player will filter out his own guess in the previous round to be able to make the best prediction of his opponents’ distribution over all numbers and then form his best response according to his level.

Ho et al. (1998) were the first to experiment with learning by repeating the game ten times. They found that there were two types of learners: adaptive learners, who best respond to experience in the previous round and sophisticated learners, who realize that other players are learning and best respond to these ‘lower-level learners’.

In my model players can learn and thus update their strategy if the winning number in the previous round is smaller (larger) than their own guess. They will however only update their strategy if their performance in the previous round according to this update would have been better than before the update.

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2.3 Equilibrium analysis

Before explaining the model that is used in this thesis, some background will be given on the equilibrium outcome of zero, so that the workings of this game are fully understood from a theoretical point of view.

2.3.1 Nash equilibrium

This equilibrium analysis will start with the Nash equilibrium of zero of the p-BCG in order to see what the rational choice would be if everyone is rational. For this proof the definitions of the Nash equilibrium and weak domination are needed and will therefore be described below. Recall that a Nash equilibrium means that once every player has chosen a strategy and the whole system is in equilibrium, no one will benefit from changing his or her strategy. Now, some definitions will be given in order to proof that the Nash equilibrium of the p-BCG is zero. Let N = {1, . . . , n} denote the finite set of players and Si the set of strategies that player i

can adopt. A strategy profile s = (s1, . . . , sn) assigns a strategy to each of the players. Each

player has a payo↵ function Fi(s) which specifies the payo↵ player i receives, given the strategy

profile s is adopted by all the players. Denote by s i a profile for the set of players N\ {i} and

Fi(s) = Fi(si, s i). A strategy s⇤i is a best response for player i against the the profile s i if

8si2 Si: Fi(s⇤i, s i) Fi(si, s i).

Definition 2.1. (Nash-Equilibrium)

A strategy profile se is an Nash equilibrium (NE) profile if for every i2 N, se

i is a best response

for player i against se_i, that is,_{8i, s}i2 Si, si 6= sei : Fi(sei, sei) Fi(si, sei).

In the p-BCG all N participants have to choose a strategy si from the interval [0, 100]. The

participant who chooses the number closest to p_{· average of all chosen numbers receives a payo↵} Fi(s) = 1. If there are several winners, the payo↵ is equally distributed over all winners.

Thus the winning strategy is the strategy si for which

|si p· Pn k=1sk n |  |sj p· Pn k=1sk n | where i_{6= j and s}i 2 [0, 100].

In the proof of the Nash equilibrium weak domination is used. Weak domination refers to the idea of a strategy profile that is at least as good as some other strategy profile. If that is the case the other strategy profile is weakly dominated.

Definition 2.2. (Weak domination)

8i, a strategy s⇤ 2 Si weakly dominates s0 2 Si if

8s i 2 S i : Fi(s⇤, s i) Fi(s0, s i)

and

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CHAPTER 2. THEORETICAL BACKGROUND 9 Following is the formal proof that the Nash equilibrium of the p-BCG game is zero. In short, this is the case because all the strategies above p· 100 are weakly dominated, since the winning number can never be above p_{· 100, and these strategies will therefore not be chosen. But then} the new interval of strategies becomes [0, p· 100]. If that is the interval then all the strategies above p_{· p · 100 are again weakly dominated. If this is infinitely repeated the interval becomes} infinitely small, which leaves us with the strategy of zero.

Proposition 2.3. The strategy se

i = 0 is a unique Nash equilibrium in the p-BCG

Proof. Since si 2 [0, 100] the upper bound maxsi{p ·

Pn k=1sk

n } = p · 100 and therefore p · 100

weakly dominates all strategies si > p· 100. We now have a new set of dominant strategies

s⇤_i 2 [0, p · 100]. But now the upper bound maxsi{p · [p ·

Pn k=1sk

n ]} = p2· 100 and therefore p2· 100

weakly dominates all strategies si > p· 100. By iterated elimination of weakly dominated

strategies we get se

i = limn!1pn· 100 = 0 is a Nash equilibrium. Let’s now say there is another

N E6= 0 : s⇤_{. Then there are two cases: either everyone has chosen the same number or at least}

one player has chosen a di↵erent number. If at least one player has chosen a di↵erent number, then there exists at least one i 2 N : Fi(si, s i) = 0 and that player can improve his guess,

thus s⇤ is not a NE. If everyone has chosen the same number then everyone can benefit from changing his guess from s⇤ to p_{· s}⇤, since at first Fi(s⇤i, s⇤i) =total payo↵/N, and afterwards

Fi(p· s⇤i, s⇤i) =total payo↵. Therefore this is also not a NE and sei = 0 is the unique Nash

equilibrium.

2.3.2 Theoretical evolution of the p-BCG

We know that zero is the rational strategy if we know that everyone in the game is rational. However it is interesting to find out what happens when players are not completely rational at first. In other words what happens to the system when players play multiple rounds and can learn from previous rounds, but are not playing zero from the start.

Therefore, as a next step, I will show what the best reply behaviour will be of a group of players who play many subsequent rounds and are initially randomly distributed over all levels l_{2 [0, m].}

Suppose that the winning player of the first round was a level-k player, meaning that the winning number was approximately pk_{· 50. Now all players know that}

8l < k : Fi(sl+1, x i) Fi(sl, x i),

where sl _{corresponds to the level l strategy. Therefore all players who choose a strategy smaller}

than k will, in the next round, choose a strategy of at least k. Because all players who choose a strategy larger than k know this, it is never a best reply to choose a level l < k in the next

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round. This implies that even though higher level players might lower their level, the average level in the second round will be higher than k. Therefore in each new round the average level becomes higher and the winning number goes towards zero. In practice however, there will be new players in the game at some point, and players can therefore not learn from their opponents infinitely many times. Therefore, we will now see what happens when some ‘irrational’ players are introduced once all the other players have reached the equilibrium.

2.3.3 Introducing ‘irrational’ newcomers

The Nash equilibrium implies that playing zero is the rational strategy. However if some players are not completely rational, or do not have enough information about other players, it might not be beneficial any more to play zero. I want to know when it is beneficial to defect from playing the rational strategy. Therefore I will now show when this happens for some cases. First, the case when some level-0 players are introduced, some level-1 players are introduced and then more generally when level-k players are introduced. Second, the case in which some players are level-k and some are level-k+1. Lastly I will show that it is always beneficial to be a level-k+1 player if some players are level-k and some are level-k+1.

First I will show the case in which some players are irrational and play the average between 0 and 100. Say q is the ratio of irrational players, playing 50. Then the average of all si becomes

¯

s = 0· (1 q)· n + 50 · q · n

n = 50q

In this case it is only beneficial to keep playing zero is p· average is closer to 0 than it is to 50, thus p· ¯s = 50pq < 1 2· 50. Thus, Fi(0) > 0, p < 1 2q.

Following the same calculation we can say that it is only beneficial to keep playing zero against a fraction of q people who play a level-1 strategy (thus p_{·50) when,}

p_{· ¯s = 50p}2q < p_·1 2 · 50. Thus again, Fi(0) > 0, p < 1 2q.

To conclude, it is beneficial to defect from playing zero if there is a fraction of q level-0 or level-1 players and p < _2q1. This can be generalized for a fraction of level-k players:

Proposition 2.4. If a fraction q plays a level-k strategy (si = pk· 50) and all other players play

zero then

Fi(0) > 0, p <

1 2q.

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CHAPTER 2. THEORETICAL BACKGROUND 11 Proof. Fi(0) > 0, p · ¯s < 1 2p k_{· 50} p_{· ¯s = p · [}0· (1 q)· n + p k_{· 50 · q · n} n ] = p k+1_{· 50q} pk+1_{· 50q <} 1 2p k_{· 50 , p <} 1 2q.

This implies that it can only be beneficial to defect from playing zero if q > p and p > 1₂ and that there are indeed cases in which the Nash equilibrium is not the best reply to one’s opponents.

Now I will show for which fraction of level-k and level-k+1 and which p, it will still be beneficial to play the equilibrium strategy of zero.

Proposition 2.5. If a fraction qk plays a level-k strategy, a fraction qk+1 plays a level-k+1

strategy and all other players play zero then

Fi(0) > 0, qk+ pqk+1 < 1 2. Proof. Fi(0) > 0, p · ¯s < 1 2p k+1_{· 50} p· ¯s = p · [0 · (1 qk qk+1) + pk· 50 · qk+ pk+1· 50 · qk+1] = pk+1· 50qk+ pk+2· 50qk+1 P· ¯s = 50pk+1qk+ 50pk+2qk+1< 1 2p k+1_{· 50} , qk+ pqk+1< 1 2.

It has bees shown that in some cases it is beneficial to defect from the Nash equilibrium. However, if a player can choose between a level-k and a level-k+1 strategy, and everyone else also chooses one of these strategies, it is always beneficial to use the highest order belief, thus level-k+1.

Proposition 2.6. s consists of a fraction q level-k players and a fraction of 1 q level-k+1 players _{) F}i(k + 1) > 0 Proof. Fi(k + 1) = 0, p · ¯s > 50· p k_{+ 50}_{· p}k+1 2 , p · ¯s = p[pk· 50q + pk+1· 50(1 q)] > 50· p k_{+ 50}_{· p}k+1 2 , q > 25· p k_{+ 25}_{· p}k+1 ₅₀_{· p}k+2 50(pk _pk+1₎ = (p + 1/2)(1 p) p(1 p) = 1 + 1 2p

But this is a contradiction since p, q2 (0, 1). Therefore Fi(k + 1)6= 0 and since a payo↵ is never

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2.3.4 Evolutionary stability of the Nash equilibrium

It has now been shown that there are indeed cases in which it is beneficial to defect from playing the Nash equilibrium strategy. In order to analyse the behaviour of the winning number, it is interesting to see whether the equilibrium is an evolutionary stable strategy (ESS). If it is an ESS, then the equilibrium will not be disturbed if there are entrants in the population who will not play the equilibrium strategy.

In the case of an infinite strategy set Oechssler and Riedel (2001) show that the discrete repli-cator dynamics can be applied without modification, as long as the payo↵ function is bounded. Therefore in this case the following definition can de used (Veelen and Spreij, 2009):

Definition 2.7 (Invasion Barrier). ✏p(Q) is an invasion barrier for P against Q, Q6= P , if

u[P, (1 ✏)P + ✏Q] > u[Q, (1 ✏)P + ✏Q] for all ✏2 (0, ✏p(Q)).

Definition 2.8 (Evolutionary Stable Strategy). P is an evolutionary stable strategy if there exists an invasion barrier ✏p(Q)2 (0, 1) for every strategy Q 6= P .

In the continuous case however it is more difficult to show that a strategy is not ESS, because of the following di↵erence (Veelen and Spreij, 2009):

In the discrete case the following holds:

ESS , IB

In the continuous case the following holds:

ESS_{( IB,} but

ESS; IB.

Therefore it can not be said that if there is no invasion barrier the strategy is not ESS.

However, in the actual case of this game, the strategy space is not infinite because a player will never choose a number with infinitely many decimals. Therefore, the definition from Weibull (1997) can be used, which is a definition for a finite strategy space but in an n-player game. Definition 2.9 (Evolutionary Stable Strategy). x2 ⇥ is evolutionary stable if for every strategy profile y_{6= x there exists some ¯✏}y 2 (0, 1) such that for all ✏ 2 (0, ¯✏y), and with w = ✏y +(1 ✏)x,

ui(xi, w i) > ui(yi, w i),

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CHAPTER 2. THEORETICAL BACKGROUND 13 I will now show that according to Definition 2.9, zero is not an ESS in the p-beauty game. Proposition 2.10. Zero is not an evolutionary stable strategy in the p-beauty game with a finite strategy set, the number of players N > 4 and p = 2₃

Proof. We want to prove that x = 0 is not an ESS. Therefore we want to prove that@✏y 2 (0, 1)

such that for all ✏_{2 (0, ✏}y), and with w = ✏y + (1 ✏)x,

ui(xi, w i) > ui(yi, w i),

for some i2 I. Or in other words: there exists some ✏y 2 (0, 1) such that for all ✏ 2 (0, ✏y), and

with w = ✏y + (1 ✏)x,

ui(xi, w i) ui(yi, w i),

for all i 2 I. Because the game is symmetric for all players we can prove it for one i. Now xi = 0, yi 2 (0, 100) and wi = ✏yi + (1 ✏)xi = ✏yi. The winning number in the case where

player i plays xi and the rest plays w i is W (xi, w i) = p· P

j6=i✏yj

N = p·

(N 1)✏yi

N . We want

to know whether this winning number lies closer to x or to w, thus whether it lies above or below the average of the two strategies: 1₂✏yi. If N = 4, W (xi, w i) = ₃2 · 3✏y₄i = 1₂✏yi, and

ui(xi, w i) > 0. But if N > 4, then W (xi, w i) = p·(N 1)✏y_N i > 1₂✏yi and ui(xi, w i) = 0. And

as long as ✏y 2 (0, 1), thus w < y, ui(yi, w i) = 0 and therefore ui(xi, w i) = ui(yi, w i) for all

i_{2 I.}

I have shown that the Nash equilibrium is not an evolutionary stable strategy. This implies that if the fraction of newcomers in this game is large enough, the equilibrium will be disturbed. Therefore it can be said that in financial markets, if the throughput is large enough, the price of an asset could move away from its fundamental value.

2.3.5 The model with newcomers

It has been shown that zero is the Nash equilibrium of the p-BCG. Moreover, I have shown that for any random initial distribution of players, everyone will eventually choose zero. However when new players are introduced this equilibrium can be disturbed, as I have shown in proving that zero is not an ESS. I want to know when this defection results in permanent non-equilibrium behaviour. Thus, when is it not beneficial anymore to choose a strategy of a higher level, because there are too many newcomers in every round. Therefore I will make a model in which players can learn, while inexperienced players are introduced after each round. The model will be further explained in the next chapter.

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The Model

In this chapter the model will be described with which the higher order beliefs system of the financial market can be analysed. For this model the p-beauty contest game was used. Apart from the obvious reason of the p-BCG being a game in which players can form higher order beliefs, there are three reasons for using this particular game. First, there are many di↵erent strategies, which gives players a lot of opportunity to gain experience. Second, it is a constant-sum-game, which means that your actions will not influence the total payo↵ of the game. This eliminates the possibility of altruistic behaviour. Finally, the e↵ect of experience is predictable and the advantage can easily be measured by the total payo↵.

Before explaining the details of the way in which the game was modelled, I will introduce some of the assumptions on the information sets available to players, the strategies which they form, the replacement of the players and the introduction of an amount of rational players.

3.1 Model assumptions

3.1.1 Information sets

When players in a game have di↵erent information sets available to them, their best reply changes. In order to analyse the e↵ect of information, the model will consist of three di↵erent model assumptions based on the information which is available and which is used by the players. In the first model the players will only use the winning number of the previous round, to form their guess. In the second model the players will also use the trend in the winning number by using the ratio of the last two winning numbers in their guess for the next round. In the third model, players are also informed about the number of newcomers which they will use in their best response to their opponents. Because the information sets become larger the e↵ects of more information on the winning number and the performance of the players can be analysed.

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CHAPTER 3. THE MODEL 15

3.1.2 Strategies

The model consists of two di↵erent strategies, namely the level-k strategy and the Cognitive Hierarchy strategy as mentioned in the previous chapter. By using two di↵erent strategies I can analyse whether the manner in which players form higher order beliefs has any influence on the winning number or their performance. This results in six di↵erent models which will be referred to as described in table 3.1.

Table 3.1: Model assumptions

Winning Number +Ratio winning numbers +Newcomers Rate

Level-k LW LR LN

CH-model CHW CHR CHN

3.1.3 Replacing experienced players

To fit the analogy to the stock market more closely, some inexperienced players need to be introduced. In my model some experienced players will be replaced by inexperienced players at random after each round and they will use the same strategy as all the players did in the first round.

3.1.4 Introducing rational players

Because results of the newspaper challenge show that there is always a fraction of players that does play the equilibrium strategy of zero, these players will be incorporated in the model. This parameter was set at 0.1, corresponding to the average of the three newspaper challenges. This also corresponds to an asset market in which some traders know the fundamental value of the asset and use this value to predict the price of the asset.

3.1.5 Parameters

In the model there are five di↵erent parameters (see Table 3.2). Some of these parameters were fixed when the results were analysed, but I will elaborate further on this in the next chapter.

3.2 Model setup

In this section I will explain the basic model first, after which the way in which the other information sets are carried out, is explained.

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Table 3.2: Parameters Number of players N

Number of rounds R

Number p of the game p Amount of learners Lr Amount of newcomers N r

3.2.1 Basic model

In order to start the game the players need to have a level of higher order belief. The initial distribution of the levels of the players is a Poisson distribution with = 1.5, which is truncated between 0 and 4, because experimental research suggests the average level in the first round of this game is 1.5, and no one will iterate more than four times (Camerer et al., 2004).

In the first round, everyone picks a number xi from a normal distribution with µ = 50· ps

1 i

(where s1_i is the assigned level from the Poisson distribution) and = 0.341· 50 · ps1

i, which is

truncated at 0 and 100 because of the interval of the game. The normal distribution is used to create variety in the chosen numbers. For the standard deviation 34.1% of the mean is used, because

P (µ _{ x  µ + ) = 0.6827.}

In order to test whether a change in this sigma had a significant influence on the first winning number, the programme was carried out ten times with sigmas of = 0.9· 50 · ps1

i and =

0.03_{· 50 · p}s1

i. An ANOVA test was used to test whether the means of the ten

first-round-winning-numbers were significantly di↵erent. The p-values of these computations were 0.109 and 0.112 respectively and therefore the null hypothesis that the means are the same cannot be rejected and the use of a sigma of = 0.341_{· 50 · p}s1

i could be used.

After every player has chosen a number, a fraction of Rr = 0.1 players’ numbers are replaced by zero, because this is the fraction of players playing the Nash equilibrium. Then the player whose chosen number (xr_i) lies closest to the winning number Wr= p·PNi=1xi

N , thus

mini|Wr xri|,

receives a payo↵ of 1.

After the game is played a fraction of Lr of players can learn from the previous round, thus Lr_{· N randomly chosen players can learn. If their number in the previous round (x}r 1_i ) lies above the winning number in the previous round (Wr 1), thus when

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CHAPTER 3. THE MODEL 17 and their distance to the winning number is larger than it would have been had they already ascended one level, thus

Wr 1 xr 1_i >_|Wr 1 p_{· x}r 1_i _|, then they ascend one level (from sr

i = k to sr+1i = k + 1). If however their number in the

previous round lies below the winning number in the previous round, thus when Wr 1 xr 1_i < 0,

and again their distance to the winning number is larger than it would have been had they already descended one level, thus

|Wr 1 xr 1_i | > |Wr 1 p 1· xr 1i |,

then they descend one level (from sr

i = k to sr+1i = k 1).

Then a fraction of N r newcomers is introduced. This happens randomly and these newcomers will again be appointed a level according to the Poisson distribution and a first guess according to the normal distribution, both described above. After the first round this procedure is repeated R 1 times.

After the first round players can use the winning number of the previous round to improve their guess. Here the di↵erence in higher order beliefs comes into place. In the level-k model, players believe that all their opponents use a strategy of one level lower than their own level, whereas in the cognitive hierarchy model players believe their opponents are Poisson distributed over all levels below their own.

Level-k model

In the level-k model, player j picks a number based on the winning number of the previous round. The player filters out his own number, to get the best prediction of its opponents, as follows. The winning number is

p_· 1 n X i xi = p· 1 n( X i6=j xi+ xj),

and therefore the winning number without one’s own guess will be (Wr 1_·N p x r 1 j )· p N 1 = (W r 1_{· N} _p_{· x}r 1 j )/(N 1). Now xr j = ps r j _{· (W}r 1_{· N} _p_{· x}r 1

j )/(N 1), where srj = k if that player is a level-k player.

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Cognitive Hierarchy model

In the Cognitive Hierarchy model players will choose a number based on the belief that all other players are Poisson distributed over all levels below his own. A level-k player’s belief about the proportion of level-h players is denoted by gk(h). We assume that level-k players do not realize

that other players might be of a higher level, thus, gk(h) = 0,8h k. Players will normalize

these proportions, such that,

gk(h) =

f (h) P

l = 0k 1f (l),

where f (k) is the probability density function of the Poisson distribution with one parameter ( ),

f (k) = e

k

k! .

For this parameter , the outcome of = 2.8 that Camerer et al. (2004) computed in a repeated experiment with only professional stock market portfolio managers, is used.

After all of the gk(h)’s are computed the Cognitive Hierarchy matrix is filled in by p· (one

players distribution on the beliefs about others). Then every player again chooses a number based on the winning number of the previous round, having filtered out one’s own guess. The rest of the model works as described above.

Performance measure

Moreover, I want to analyse whether experience and di↵erence in information sets will improve performance. Therefore, a performance measure was constructed. This performance measure (Pm) was based on the relative distance of a player’s guess to the winning number:

P mr_i = P mr 1_i +_|1 W

r

xr i |.

This measure can be compared to the amount of times a player is replaced during the entire game.

3.2.2 Ratio of winning numbers

Above the basic model is described. Now I will show how the extra information available to the players, is incorporated in the model.

If a player wants to play his best response to his opponents, he needs to know their level. The distribution of the levels of a player’s opponents is not included in one’s information set. However a player can use the ratio of winning numbers of the previous two rounds to form a more accurate belief about the level of his opponents and the trend followed by the winning number. Therefore, in this model, the ratio

Wr 1 Wr 2

is used in the prediction for a players guess making xr_j = W r 1 Wr 2 · p sr j _{· (W}r 1_{· N} _p_{· x}r 1 j )/(N 1).

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CHAPTER 3. THE MODEL 19

3.2.3 Information of newcomers rate

If players know how many newcomers are introduced in each round, they can use this infor-mation to form a best response that includes their belief on the behaviour of these newcomers. Players believe that a newcomer will iterate on average 1.5 times (as they themselves did in the first round), therefore making their guess p1.5· 50. The experienced players will filter out the replaced players from the winning number and replace it with their belief about the guess of the newcomers resulting in their own guess being

xr_i = W r 1 Wr 2 · p sr i · (Wr 1· (N N · Nr) p· xr 1 k + p1.5· 50 · N · Nr)/(N 1).

3.3 Estimation

In my model there are several variables that will influence the speed of convergence towards the Nash equilibrium. A lower p, a lower newcomers rate and a higher learning rate will all contribute to a higher speed of convergence. The introduction of the players’ ability to use the trend of the winning number in their prediction for the next round, will induce more volatility. The model will converge to the Nash equilibrium if there are no newcomers, even when the learning rate is zero. This is because, if every player remains a level-0 player and thus always chooses the winning number of the previous game, the winning number of each subsequent game will be (2/3) of the winning number of the previous game.

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Results

In this chapter the results of the model will be described. The results are divided in the path of the winning number and the performance of the players, for all di↵erent models. Before elaborating on the results, the reason why some of the parameters are fixed, is explained.

4.1 Parameter choice

There are five di↵erent parameters in all six models. In order to analyse the results, some parameters are fixed. This paragraph will elaborate on why these parameters were chosen. In table 4.1 the fixed parameters are shown.

Table 4.1: parameter choice

Parameter Value Explanation

N 10 Number of players

R 300 Number of rounds

p 2₃ Number p of the game

Lr 1 Learning rate

4.1.1 Number of players

The number of players in a game influences its equilibrium behaviour, because the percentage of newcomers in a round in which there is one newcomer varies with the amount of players. For instance if N = 10, then in a round in which a player is replaced, 10% of the players are replaced, whereas if N = 100, only 1% is replaced. However, since this di↵erence is dependent on the newcomers rate, which is another variable, a number N can be chosen which is thought to be reasonable. Therefore the number of players was chosen for which one’s own guess significantly influences the aggregate behaviour. For this study the number of N = 10 was chosen.

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CHAPTER 4. RESULTS 21

4.1.2 Number of rounds

Within 300 rounds, the winning number always converged towards one number around which it oscillated. This was almost always the case within 5 rounds. However, in order to be certain that nothing significant happened after 20 rounds, 300 rounds were used for the first result. In order to analyse the oscillation of the winning number more closely, 30 rounds were used for later results.

4.1.3 Number of p of the game

In almost all of the experimental literature of the p-beauty contest game the number 2/3 is used as the p. To be able to use the outcome of this literature (for instance for the choice of in the CH-model) I chose to use the same p. In all models convergence is not dependent on the choice of p. The speed of convergence, however, is. For instance, in the basic level-k model the winning numbers in the 4th round are displayed in table 4.2

Table 4.2: Winning number in the fourth round for di↵erent p’s, Lr=1, Nr=0.1 and Rr=0.1

p Winning number

0.3 W ⇡ 0.3 2/3 W ⇡ 1.3

0.9 W ⇡ 15

4.1.4 Learning Rate

Because the average level stays roughly the same in all models, the learning rate has very little influence on the winning number. Therefore a learning rate of 1 was chosen, corresponding to a case in which every player is able to learn in every round. Moreover, this is in accordance with financial markets in which traders are able to learn after every transaction.

4.1.5 Depth of reasoning

Experimental studies show that players in strategic games use some level of depth of reasoning. In this game that will be the level of higher order belief, or the level k. There is, however, an upper bound to this depth of reasoning, because humans can not compute in the same manner that computers can. Ohtsubo and Rapoport (2006) show that, in a beauty contest, most people only use a depth of reasoning of two levels and very few use higher levels of 3 or 4. Therefore, in this model an upper bound to the level of higher order beliefs of 4 is imposed.

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4.2 Results of the model

This section will start with the results of the winning number of the game of the level-k model followed by the CH-model. Then the results of the performance measure will be shown.

4.2.1 Winning number

Level-k model

The winning number stops going to zero even if there is only one newcomer every round (N r = 0.1). It can be seen in figure 4.1, that the winning number oscillates around 3. The winning number goes towards zero very quickly, but after just 4 rounds it starts moving upwards again. This implies that some sort of new equilibrium is formed, which is a deviation from the Nash equilibrium. The average level stays very low. Apparently it is not beneficial to use higher order beliefs of more than two iterations.

Figure 4.1: Plot of winning numbers and average levels of the LW-model

If the newcomers rate becomes higher, then so does the winning number. For instance, with a newcomers rate of 0.5 the winning number is on average 12 and never drops below 7. However when the newcomers rate is 0.01, meaning that only one person is replaced in a game with 100 players, the winning number is on average 0.5, but it never drops below 0.1 and thus never reaches the equilibrium of zero. Therefore, the amount of newcomers has a large influence on the deviation from the Nash equilibrium.

The di↵erence in information sets creates di↵erent results. In order to see the di↵erence between the LW, LR and LN models the results are shown for the first 30 rounds. As can be seen in

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CHAPTER 4. RESULTS 23 figure 4.2 the changes in the winning number are small and a deviation from the average never lasts longer than a few rounds. Moreover, it can be seen in figure 4.3 that the deviations become larger when players use the trend of the previous winning numbers. In figure 4.4 it is shown that if players know how many newcomers there are in each round the deviations become even larger. This means that more information leads to a higher volatility.

Figure 4.2: Plot of winning numbers and average levels of the LW-model

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Figure 4.4: Plot of winning numbers and average levels of the LN-model

In order to test the di↵erence between these models, the programme was run for 10.000 rounds. The mean and standard deviation can be seen in table 4.3. Here we see that from the LW to the LN model the standard deviation is almost doubled. The most significant di↵erence is caused by using the trend of the winning numbers, thus from the LW to the LR model. Furthermore, table 4.3 shows that the average of the winning numbers is much larger in the LN model than in the LW model, implying that more information leads to a larger deviation from the Nash equilibrium of zero.

Table 4.3: Mean and standard deviation of the three di↵erent level-k models with Lr=1, Nr=0.1 and Rr=0.1

µ

LW 3.1 1.2

LR 3.5 1.8

LN 4.4 2.1

In order to test whether these means are significantly di↵erent an ANOVA test was used. First the H0 : µLW = µLR was tested. The p-value was 6.26e 77, so the null hypothesis that the

means are the same can be rejected. Then the same was done for H0 : µLR = µLN, and again

the p-value was very small (5.27e 256), so that the null hypothesis could be rejected and all the means are significantly di↵erent. The boxplots of these computations can be seen in figure 4.5.

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Figure 4.5: Boxplots of the winning numbers of the three models over 10.000 rounds A two sample F-test was used to test whether the variances come from the same distribution. The null hypothesis was again rejected with both the comparison between LW and LR, and between LR and LN, with p-values respectively 8.68e 232 and 1.68e 122, implying that the variances are indeed significantly di↵erent and more information leads to a higher volatility.

CH-model

In order to see whether the di↵erence in the way players make higher order beliefs was significant a comparison of the level-k and CH-model was made. There is no significant di↵erence in the behaviour of the winning number between the level-k and the CH-model. The similarity between the two models can be seen in figure 4.6 in comparison to figure 4.1.

The two models do not significantly di↵er in their standard deviation of the winning number. With a newcomers rate of 0.5 and a 10.000 times repeated game, the of the level-k and CH-model were 2.583 and 2.590 respectively (a larger newcomers rate was used because the deviations then become larger and the comparison of the standard deviation is therefore more accurate). The similarity of the variances was tested using a two-sample F-test with the null hypothesis that the variances come from the same distribution. The null hypothesis could not be rejected with a p of 0.4943, implying that the variances are indeed not significantly di↵erent. Moreover, the mean and standard deviation of the three di↵erent CH-models with a newcomers rate of 0.1 were all the same, when rounded in one decimal, as in table 4.3. This is due to the fact that the di↵erences in these models lie in the di↵erence on the beliefs about the levels of the opponents. However, since the average over all levels remains approximately 2, the di↵erence

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Figure 4.6: Plot of winning numbers and average levels of the CHW-model

between the belief that all other players are of level 1 (level-k model) and the belief that all other players are distributed over level 0 and 1, is very small.

In order to compare the two models, the level was manually fixed at a higher level using the Poisson distribution with = 5 truncated at (0, 10) and a learning rate of 0. The resulting means from a 10.000 times repeated game were µLW = 3.38 and µCHW = 3.60, which were

significantly di↵erent using an ANOVA test with a p-value of 2.02e 29. The average winning number is slightly higher in the CH-model, due to the underestimation of the opponents. The variances however, were still not significantly di↵erent (using an F-test with p-value 0.9820), implying that the beliefs on the levels of the opponents does not influence the path of the winning number.

4.2.2 Performance measure

In order to see whether players who were replaced less often performed better in the long run the game was repeated 300 times. In figure 4.7 it can be seen that in the basic level-k model, players who were replaced most frequently on average had a much larger deviation from the winning number.

Because there are only 10 players, thus few observations, the bootstrap method was used to test whether there was a positive correlation between performance and experience. 1000 bootstrap data samples were used to calculate the variation of the correlation coefficient. This can be seen in figure 4.8. The sample minimum was positive, indicating that there is indeed a positive relationship between the performance measure and the number of times a player is replaced. The standard error of the bootstrap correlation coefficient was only 0.0734. Results from the LR, LW and CH models are similar. Therefore, it can be concluded that more experience leads to a better performance.

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Figure 4.7: Plot of the performance of all 10 players and the number of times they are replaced over 300 rounds in the basic model

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By using this performance measure it can also be derived in which model players performed better, thus in which model the players on average made the best prediction for the next round. In order to analyse this, the average performance measure over the ten players for 10.000 rounds, was computed and shown in table 4.4. These numbers show that by using the trend, the performance actually becomes worse. However, if a player knows the number of newcomers introduced after each round, his prediction becomes much better. This means that even though the knowledge of the newcomers ratio leads to much more fluctuation in the winning number, the prediction of that winning number becomes better. Table 4.4 also shows that players in the CH-model perform better in all three models. These di↵erences in the mean were again tested by using an AONVA test. The p’s for the LW, LR and LN test were respectively 5.96e 17, 3.23e 14 _{and 8.27e} 20_{, thereby rejecting the null hypothesis that the means of the level-k and}

CH-models are the same.

In short, using the trend worsens the performance but having information on the newcomers rate improves it and players using the CH strategy had a better performance than players using the level-k strategy.

Table 4.4: Average performance measure of the three di↵erent models with Lr=1, Nr=0.1 and Rr=0.1

µ of Level-k µ of CH LW 1.35_{· 10}4 _1.08_{· 10}4

LR 2.21_{· 10}4 _1.64_{· 10}4

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Chapter 5

Conclusion

Traders in financial markets have to form higher order beliefs: success in these markets may depend on how well traders anticipate the beliefs of other traders and the beliefs of these other traders on yet other traders. Therefore, experience and thereby knowledge of the behaviour of fellow traders can be important. Research suggests that the loss of experience can, among other things, lead to bubbles and crashes.

This research aimed to investigate the influence of inexperienced employees in financial markets. In order to research this, the p-beauty contest game, a game in which players have to form higher order beliefs, was used. In this game a winning number is computed which can be related to the price path of an asset.

The research entailed a theoretical equilibrium analysis of the game and the development of a programmed model to analyse the behaviour of the winning number of the p-BCG and the per-formance of the players, when the game is frequently repeated, players can learn and newcomers are introduced. Three di↵erent models were made, that di↵er in the amount of knowledge that is used by the players.

5.1 Theoretical equilibrium

In order to fully understand the theoretical workings of the p-BCG, a theoretical equilibrium analysis was carried out. The Nash equilibrium of the p-beauty contest game is zero. However, when there is a fraction q of players playing a level-k strategy and p > _2q1 it is beneficial to defect from playing the Nash equilibrium. Moreover, if the strategy space is assumed to be finite, the Nash equilibrium is not an evolutionary stable strategy. Therefore, if there are enough newcomers, who do not play the NE of zero, the equilibrium will be disturbed. This means that if there are enough traders who would not pay the fundamental price of an asset, the best response is to also deviate from the fundamental price.

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5.2 Winning number

The model of the p-BCG induces a path of the winning number over the course of all the rounds for which the game is played. Three things could be concluded from the path of the winning number.

Firstly, even when there is a very small fraction of newcomers in the repeated p-BCG the winning number stops going towards the NE of zero. A new equilibrium is formed, around which the winning number oscillates, which is dependent on the fraction of newcomers. This implies that, in financial markets, the price of an asset will deviate from the fundamental price even if there are very few new employees. The size of this deviation heavily depends on the amount of newcomers during the time in which traders update their belief on the price of the asset. A higher newcomers rate can lead to very large deviations from the fundamental price and might induce bubbles and crashes.

Secondly, across the three models the variance of the winning number was di↵erent. In the basic model the standard deviation was only 1.2. Once the extra information on the trend of the winning number was used in the players’ guess, the standard deviation grew to 1.8 and if players could use the information of the newcomers rate it grew to 2.1, almost double as volatile as the basic model. This result implies that if traders have or use more information the volatility of the asset will increase. It was expected that more volatility could be induced by using the trend, but not by information on newcomers. More public information was expected to lead to the price moving towards the fundamental price and not the other way around. Finally, the path of the winning number is neither dependent on the beliefs of players about the level of its opponents nor on the amount of iterations in the higher order beliefs. This implies that the volatility and value of the asset are not influenced by how many iterations stock brokers make or believe that others make.

5.3 Performance

A performance measure was built into the model, from which the di↵erences between the models could be evaluated in relation to the performance of the players.

The performance measure has three interesting results. Firstly, if a player is replaced more, and thus has less experience, his performance measure is significantly higher (a higher performance measure means a larger deviation of a player’s guess from the winning number). This means that an experienced trader, as expected, will on average perform better, and thus earn more. Therefore, apart from more experience leading to less bubbles and crashes, it is also beneficial for the financial institutions to hold on to their experienced traders, because they will earn more.

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CHAPTER 5. CONCLUSION 31 Secondly, even though more information leads to more volatility of the winning number, it also leads to a better average performance. If a player has information on the fraction of newcomers, it’s guess is closer to the winning number than if it does not have this information. Therefore, if a trader knows the average throughput of employees in his asset market, he can significantly improve his estimation of the price of the asset, even though it is more volatile.

Lastly, making use of the knowledge of the past of the winning number does not lead to a better performance. The winning number is such a volatile number that using the trend will on average not be beneficial. This is in accordance to the pricing strategy of an asset, in which information of the former trend is never used as an explanatory variable for the estimate of the future.

5.4 Future research

This thesis aimed to provide a contribution to the literature on the beauty contest game as an analogy for financial markets. For future research the model that was used could be expanded and experimental evidence that supports the theoretical model in this thesis should be carried out.

First of all, the model described in this thesis could be expanded by adding a variable newcomers rate in order to more closely correspond to the throughput on financial markets, which is never constant and more likely to go in phases. Moreover, the consequences of even more information could be tested. Players could, for instance, be aware of the true distribution of the levels of higher order beliefs of their opponents. This is however rather unlikely to be the case in the applications of the p-BCG.

Furthermore, the p-BCG can be experimentally carried out in the same manner as it is done in this thesis. The p-BCG has been carried out in multiple rounds before, but never for more than ten rounds. The theory that the winning number goes towards zero with no newcomers should be experimentally investigated. Another repeated p-BCG experiment could be carried out, in which there are newcomers. From this experiment the actual reaction of experienced players to inexperienced players can be analysed.

Lastly, a real life asset market experiment could be carried out. If there are two experiments, with and without newcomers, the di↵erences can be analysed. It should be tested whether the price of an asset is indeed further away from the fundamental price if there are inexperienced players and whether the more experienced players earn more on average. In this experiment it can also be tested whether more information leads to a better performance, by using another test group that is aware of the number of newcomers.

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In this chapter the Matlab codes for the models are shown. The first Matlab code is the entire basic level-k model. Thereafter, parts of the Matlab codes are given in which the di↵erences between the models can be seen.

5.5 Basic model

function [W,F,s]=pbeauty1(N,R,p,Lr,Nr,Rr) W=zeros(1,R); %vector of winning numbers F=zeros(N,R); %payoff matrix

Fabs=zeros(N,R); %performance measure s=zeros(N,R); %levels

x=zeros(N,R); %matrix of chosen numbers

Distance=zeros(N,R); %distance to the winning number Newcomers=round(Nr*N); %amount of newcomres in each round Rational=round(Rr*N); %amount of rational players

Newcomersmatrix=zeros(Newcomers,R); %which players are replaced %THE GAME

for r=1:R;

if r==1; %first round

Poisson = makedist(’Poisson’,’lambda’,1.5);

tP = truncate(Poisson,0,4); %truncate Poisson distribution for n=1:N;

s(n,1)=random(tP);

Normal = makedist(’Normal’,50*p^(s(n,1)),(50*p^(s(n,1))*0.341)); %create normal distribution for first round

tN = truncate(Normal,0,100); %truncate normal distribution x(n,1)=random(tN); %every player picks a number

end

W(1,1)=p*(sum(x(:,1))/N); %winning number in the first round else

for n=1:N; if r==2;

x(n,r)=p.^s(n,r)*(W(1,r-1)*N-x(n,r-1)*p)/(N-1); %filter out one’s own guess %every player picks a number according to his level and based on the

winning number in the previous round else

x(n,r)=p.^s(n,r)*(W(1,r-1)*N-x(n,r-1)*p)/(N-1); 34

(40)

APPENDIX 35 end

end end

%RATIONAL PLAYERS

RandRrN=randperm(N,Rational)’; %randomly chosen players who will be rational for n=1:Rational;

x(RandRrN(n),r)=0; end

%NEWCOMERS

RandNr= randperm(N,Newcomers); %randomly chosen players who will be dismissed Newcomersmatrix(:,r)=RandNr’;

for d=1:Newcomers;

s(RandNr(d),r)=random(tP); %then a random player will be relpaced with a level from the truncated Poisson distribution

Normal = makedist(’Normal’,50*p^(s(RandNr(d),r)),(50*p^(s(RandNr(d),r))*0.341)); %create normal distribution for first round

tN = truncate(Normal,0,100); %truncate normal distribution x(RandNr(d),r)=random(tN); %every player picks a number end

%CONTINUATION GAME if r>1;

W(1,r)=p*(sum(x(:,r))/N); %the winning number end

for n=1:N;

Distance(n,r)= x(n,r)-W(1,r); %distance to the winning number end

absDistance=abs(Distance); %absolute distance to the winning number

[~,I]=min(absDistance(:,r)); %M (~) is vector of minimum values of winning number in each round,I is vector of row indices of that number

if r>1;

F(:,r)=F(:,(r-1)); %take winnings of last round to this round end

F(I,r)=F(I,r)+1; %winner receives payoff of 1

%LEARNING

s(:,r+1)=s(:,r); %repeat the same level in the next round LrN=round(Lr*N);%amount of learners

RandLrN=randperm(N,LrN)’; %randomly chosen players who will learn RandLrDistance=Distance(RandLrN,r); %distance of learners

for n=1:LrN;

if RandLrDistance(n)>0 && RandLrDistance(n)>abs(W(1,r)-p*x(RandLrN(n),r)); %if a player’s guess is higher than the

winning number and it is chosen to learn s(RandLrN(n),r+1)=s(RandLrN(n),r)+1; end

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end

for n=1:LrN;

if RandLrDistance(n)<0 && s(RandLrN(n),r)>1 &&

abs(RandLrDistance(n))>abs(W(1,r)-x(RandLrN(n),r)/p);

%if a player’s guess is lower than the winning number and it is chosen to learn s(RandLrN(n),r+1)=s(RandLrN(n),r)-1;

end end

for n=1:N; %research suggests no one iterates more than four times if s(n,r+1)>4;

s(n,r+1)=4; end

end

for n=1:N; %create performance measure

if x(n,r)==0 && r>1; %if rational nothing happens Fabs(n,r)=Fabs(n,r-1);

elseif x(n,r)==0; %if rational in first round get one Fabs(n,r)=1; elseif r==1; Fabs(n,r)=abs(1-W(1,r)/x(n,r)); else Fabs(n,r)=Fabs(n,r-1)+abs(1-W(1,r)/x(n,r)); end end end Timesnew=zeros(N,1); for n=1:N

Timesnew(n,1)=numel(find(Newcomersmatrix==n));%how many times is each player replaced end %PLOTTING xaxis=1:N; Fabsplot=Fabs(:,R)’; [ax,~,p]=plotyy(xaxis,Timesnew,xaxis,Fabsplot,’bar’,’plot’); title(’Performance and experience’)

xlabel(’Player’)

ylabel(ax(1),’Number of times replaced’) ylabel(ax(2),’Performance measure’) p.LineWidth = 3; p.Color = [0,0.7,0.7]; one=ones(R,1); R=cumsum(one); S=sum(s)./N; one2=ones(length(S),1); Rs=cumsum(one2);

Higher order beliefs in the financial market : equilibrium of winning number and performance analysis in a repeated p-beauty contest game