Master’s Thesis
The Dynamics of Costly Signalling: the Evolution of
Cooperative Communication
Stijn Reijne
Student number: 10197109 Date of final version: July 31, 2016 Master’s programme: Econometrics
Specialisation: Mathematical Economics Supervisor: M. J. van der Leij
Second reader: T. A. Makarewicz
Abstract
Costly signalling theory provides a way to stimulate cooperation between players in a signalling game, however the consequences of costly signalling differ for each signalling game. This thesis explores the dynamics of an adapted version of the used car game (Akerlof (1970)), where it is shown that costly signalling can actually decrease the amount of information sharing between the players. The partially communicative hybrid equilibrium plays an important role in this decrease of information sharing, while this equilibrium is ‘only’ a mixed equilibrium.
i This document is written by Student Stijn Reijne who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
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1 Introduction ... 1
2 The Models ... 4
2.1 The Market of Used Cars ... 4
2.2 Adapted Used Car Game ... 6
2.3 Equilibria of the Used Car Games ... 7
2.3.1 Weak Sequential Equilibria of the Original Used Car Game ... 8
2.3.2 Nash Equilibria of the Adapted Used Car Game ... 10
2.4 Dynamics of the Used Car Games... 10
2.5 Dynamical Stability Concepts ... 12
3 Results ... 13
3.1 Dynamical Consequences of Costly Signalling ... 13
3.2 Dynamical Stability of the Sequential Equilibria ... 14
3.3 Basins of Attractions of the Original Used Car Game ... 16
3.3.1 Repairing Costs below 1 ... 16
3.3.2 Repairing Costs of 1 ... 18
3.3.3 Repairing Costs between 1 and 3 ... 18
3.3.4 Repairing Costs of 3 ... 19
3.3.5 Repairing Costs of Higher than 3 ... 20
3.3.6 Expected Payoffs of the Players ... 21
4 Discussion ... 23 5 Conclusion ... 25 6 References ... 27 7 Appendices ... 28 7.1 Appendix A ... 28 7.2 Appendix B ... 32 7.3 Appendix C ... 33 7.4 Appendix D ... 36
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Chapter 1
Introduction
Transferring information is a basic feature of life that includes signalling between humans or between animals. However, you don’t know whether the one sending the signal is telling the truth or is lying. A way to analyse this information sharing feature is to make use of game theory. In game theory there exist a model that is called a signalling game. The simplest signalling games model interactions between two individuals; a sender and a receiver. It begins with a state of the world that is determined by Nature with a probability function. The sender acquires private information about the state of the world. Based on the information of the state of the world the sender chooses his signal. The receiver doesn’t acquire the information of the state of the world and only acquires the signal that has been sent by the sender. Based on this signal the receiver chooses his action. So the eventually outcome of the signalling game depends on the state of the world, the action chosen by the receiver and (possibly) the signal sent by the sender.
Furthermore, the interests of the sender and the receiver may be aligned, partially opposed or totally opposed. If their interests are aligned they like the same outcomes and they both profit if they cooperate. If their interests are partially opposed they sometimes prefer the same outcomes and sometimes not, so then it is sometimes best to cooperate and sometimes not. Finally if their interests are totally opposed the sender and the receiver never prefer the same outcomes, hence then it is never profitable for each player to work together. These different types of interests impose different signalling games with different potential sequential equilibria. The scenario when the interests of the sender and receiver are fully aligned was first introduced by Lewis (1969). In Lewis’ signalling games there is exactly one act that is ‘right’ for each state of the world. If in each state of the world the ‘right’ act is chosen by the sender and the receiver, this results in a payoff of 1 for both players. If another act is chosen by the sender or the receiver, they both get a payoff of 0.
An example of a more opposed interests game is the Spence game (1973). Spence analysed the job market by a signalling game. In the job market, employers would like to hire only highly qualified job candidates and no low qualified job candidates. But the level of the job candidate is not directly observable. So, in order to get an indication of the abilities of the employee, a job candidate can send a signal about her qualification to the employer. But it is easy to understand that an employee will always choose to send that she is high qualified even when she is not. This leads in some cases to false signalling, because of partially opposed interests.
2 Another example of a partially opposed interests game is the Sir Philip game (1991) used in biology. In the Sir Philip game there are two players, a child and a parent. The child is the sender and the parent is the receiver. The two states of the world are that the child needs to be fed or that the child doesn’t need to be fed. The sender can send the signal ‘needy’ or ‘not needy’ and would like to be fed in either state of the world. The receiver can choose between feeding the sender and not feeding the sender and would only like to feed the sender when she is needy. If the sender is not needy the receiver would prefer to eat herself. Like the Spence game this creates a partially opposed interests game between the sender and the receiver. So, this probably leads to false signalling as well.
A way to deal with this false signalling is to introduce a costly signal. In this case there is no incentive to lie if the costs of the signal are high enough. In the Spence game costly signalling leads to low qualified job candidates not being able to afford to send a false signal and only the high qualified job candidates being able to send the high qualified signal. Also in the Sir Philip game, if costly signalling is introduced, the sender (child) doesn’t send the signal ‘needy’ when she is not, if the costs of the signal ‘needy’ is high enough. This leads to honest signalling where their interests realign and this means that costly signalling can be used to influence the potential equilibria.
The usual way to analyse the behaviour of signalling games is to search for Nash or sequential equilibria, but this requires rational players. Rationality however is a strong assumption that can be violated in practice. So what if players don’t have common knowledge about rationality? In this case you can look at a refinement of the Nash equilibrium namely Evolutionary Stable strategies. Evolutionary Stable strategies are robust against mutations in the sense that if there are some players that act differently from the rest of the population it is still best to play that particular strategy. Nash equilibria and Evolutionary Stable strategies are both static concepts, so if there are two or more equilibria in a game you don’t know which one is more likely to be reached. Hence, an alternative of these concepts is to make use of an evolutionary dynamics model.
This thesis will make use of evolutionary dynamics and stability concepts to analyse a signalling game, where interests of the sender and receiver are aligned and more opposed. The signalling game that will be analysed is an adapted version of the used car game of Akerlof (1970). Costly signalling will be used as a tool to change the game from an opposed interests game into an aligned interests game. The evolutionary dynamics model that will be used is a two-population replicator dynamics model, which can be used for asymmetric signalling games. By using the replicator dynamics model and stability concepts, a precise analysis will be given about which sequential equilibrium will be reached in the used car game for different costly signals. By doing so, answers are tried to be found on the questions: how and why do the dynamics change for different costly signals in this particular game and, more general, could increasing costs of a signal actually decrease the amount of cooperation between the players?
It is interesting to investigate the adapted used car game by evolutionary dynamics, because lately it has become more popular to analyse signalling games by evolutionary
3 dynamics models instead of only sequential equilibria. This game is, as far as I know, not analysed yet with an evolutionary dynamics model, so it might give new insights into the game and in the consequences of costly signalling in general.
There are already some games analysed using evolutionary dynamics. Hutteger, Skyrms, Smead and Zollman (2010) analysed the evolution of Lewis signalling games, where the states occurred with equal probability, using replicator dynamics. As stated before, Lewis signalling games are games with common interests of the sender and the receiver. Hutteger, Skyrms, Smead and Zollman (2010) found that, for 2 states, 2 signals and 2 acts, dynamics converge to the only separating equilibrium. This is called perfect information transmission. But when the number of states, signals and acts exceed 2 the replicator dynamics sometimes converge to the separating equilibrium and sometimes converge to a partial pooling equilibrium. This is called partial information transmission.
Hutteger and Zollman (2010) analysed a partially opposed interests game namely: the Sir Philip Sidney game. They show that the costly signalling equilibrium and certain states of no communication are stable under the replicator dynamics. Besides, they also show that a hybrid equilibrium, an equilibrium where the sender mixes his strategies, could exist and could be stable. While this hybrid equilibrium is not evolutionary stable, it is only Lyapunov stable, it turns out that this equilibrium is still of great importance. In order to see this, they look at the basins of attractions and they display that the initial points converge just as often to the hybrid as to the separating equilibrium, while the separating equilibrium is evolutionary stable.
Wagner (2012) investigated the evolutionary dynamics of zero-sum signalling games with free signals. A zero-sum game is with totally opposed interests of the players. Hence in a zero-sum game there are no separating equilibria possible. Expected was that there would be no transmission of information of the sender to the receiver, as that was established on research based on static concepts such as Nash equilibria and sequential equilibria. But, Wagner found that there is still some information transfer off equilibrium of varying degrees, because there exists a strange attractor in the interior of the state space. So he came to the conclusion that adaptive dynamics can enable information transmission even though messages at equilibria are meaningless.
The rest of the thesis is organized as followed. Section 2 discusses the signalling models and the replicator dynamics models. Section 3 analyses the results of the models. Section 4 contains the discussion and Section 5 consists of the conclusion.
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Chapter 2
The Models
This chapter illustrates the models that are used to examine the differences in dynamics of a two player signalling game with opposed interests and aligned interests of the sender and the receiver. First, the original game is introduced and transferred from opposed to aligned interests game by costly signalling. Then an adapted used car game is introduced to clarify the information sharing behaviour of the players. Thereafter, the equilibria of the games are analysed. Finally, the evolutionary dynamics models will be discussed and some evolutionary stability properties are considered.
2.1 The Market of Used Cars
Akerlof (1970) examined how quality of goods traded in a market can degrade in the presence of information asymmetry between buyers and sellers. Akerlof used the market of used cars as an example to explore this phenomenon. This market can be explained using a signalling game. In this thesis the used car game is used as an example to show the differences in dynamics of different interests signalling games, hence the game will not be exactly the same as the game of Akerlof. This means that assumptions and the corresponding payoffs of the signalling games in this thesis are chosen arbitrarily. For simplicity it is assumed that there are two types of used cars, namely: good cars and bad cars (which are notated as ‘lemons’ in the paper of Akerlof). The game can be summarized as follows:
The quality of the car is determined by Nature, with the probability of a high quality car equal to q and the probability of a low quality car equal to (1 – q).
The seller of the used car (sender) has private information about the quality of the car, so he knows whether the quality of the car is either high or low. The potential buyer (receiver) hasn’t got this information and only knows the probabilities of the car being good (q) or bad (1 – q).
The seller sends a signal which is revealed to the potential buyer, this signal is either a high price or a low price.
The receiver chooses an action and this action may depend on the observed signal.
The utilities of the sender and the receiver may depend on the quality of the car, the sender’s signal and the receiver’s action.
5 From now on the seller is mentioned as sender and the potential buyer as receiver. Let’s assume that the receiver of this game is always happy with a high quality car, unhappy with an expensive low quality car and slightly prefers not to trade a low quality car for a low price. The sender prefers to sell the car for a high price for any quality. Hence the interests of the players in this game are opposed, because the sender prefers a high price and the receiver prefers a low price and the players will probably not cooperate with each other.
To analyse the difference between opposed interests and aligned interests signalling games, the used car game need to be adjusted such that the interests of the players becomes more aligned. This can be done by making use of costly signalling. Costly signalling means that when the sender sends a particular signal he has to pay something in terms of payoff to send this signal. As briefly mentioned in previous paragraph the signal of a high price is dominant for the sender, so in order to make this signal less interesting repairing costs (c) are introduced. Let’s assume that if the sender asks a high price and it turns out that the car breaks down in a short time period that the sender has to pay the repairing costs. Another assumption is that a low quality car always breaks down and a high quality car never breaks down. So, these assumptions give the receiver some insurance that the sender will probably tell the truth more often when c increases. Notice that c could only be positive, otherwise it doesn’t make sense. The extensive form of the used car game is displayed in Figure 1 with the corresponding normal form game in Figure 2. For simplicity both quality types occur with equal probability, so q is equal to 0.5.
Figure 1. The extensive form representation of the original used car game with the type of the
car determined by Nature. The sender can only send two signals: low price and high price. The receiver can either buy the car (trade) or don’t buy the car (abstain). Player 1 is the sender and player 2 is the receiver. The first component of the payoff vector is player 1’s payoff and the second component is player 2’s payoff and c is the amount of repairing costs. For the game without insurance the repairing costs equal 0.
2 Trade Trade Abstain Abstain Low price Low price 0.5 1 Nature 0.5 1 High price High price 2 Abstain Trade Abstain Trade (3 , 3) (2 , 1) (3-c , -2+c) (0 , 1) (1 , 5) (2 , 1) (0 , 1) (2 , 0.5) (High quality) (Low quality)
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Figure 2. The normal form representation of the original used car game (q = 0.5). First
element of the payoff matrix corresponds to the sender and the second element corresponds to the receiver. Best responses (if determined) are denoted with thick figures. For the game without insurance the repairing costs equal 0.
Sender \ Receiver Trade, Trade Trade, Abstain Abstain, Trade Abstain, Abstain High, High 3-0.5c, 0.5+0.5c 3-0.5c, 0.5+0.5c 1, 1 1, 1
High, Low 2.5, 1.75 1.5, 2 2, 0.75 1, 1
Low, High 2-0.5c, 1.5+0.5c 2.5-0.5c, -0.5+0.5c 0.5, 3 1, 1
Low, Low 1.5, 2.75 1, 1 1.5, 2.75 1, 1
2.2 Adapted Used Car Game
Before looking at the equilibria of the game, the used car game is somewhat adapted. The reason for this adapted version is because the point of interests of this thesis doesn’t only lie in the outcomes of the game, but also in the information sharing feature of the players. This becomes more clear if the game is somewhat adjusted into strategies that correspond directly into perfectly sharing information and not sharing information at all. Hence, once again, the only purpose of this game is to clarify the consequences of costly signalling. The adaption can be done in the following manner.
It is natural to focus on three pure strategies for the sender, namely: always sending a high price, always sending a low price or send a high price when the quality of the car is high and sending a low price when the quality of the car is low. Let’s call these three pure
strategies High, Low and Sep. The fourth strategy of the sender that sends a low price when the quality of the car is high and a high price when the quality of the car is low will be removed because it is irrelevant. The other preferences of the players stay the same as in the original used car game.
To focus on the information sharing feature the receiver can either act as the signal carries no information at all in which case he is pooling or act as if the signal perfectly
identifies the sender’s type in which case he chooses Trade when the signal is a high price and
Abstain when the signal is a low price. The former strategy is called Pool and the latter strategy is called Sep. If the receiver plays Pool he doesn’t give meaning to the sending signal so let’s assume then that he plays Trade with 50% and Abstain with 50%, hence the payoff of
Pool is the expected payoffs of this strategy. The quasi-extensive form game is shown in
Figure 31. Figure 4 illustrates the corresponding 3 x 2 normal form game, where the payoffs are the expected payoffs of the quasi-extensive form game.
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Figure 3 is called a quasi-extensive form game instead of an extensive form game, because the receiver got only two strategies (Pool and Sep) instead of four (which would be the number of strategies in the extensive form game).
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Figure 3. The quasi-extensive form representation of the adapted used car game. The type of
the car is determined by Nature. The sender can only send two signals: low price and high price. The receiver can either act as the signal perfectly indicates the sender’s type or as the signal doesn’t carry any information of the sender’s type. The first component of the payoff vector is player 1’s payoff and the second component is player 2’s payoff and c is the amount of repairing costs. For the game without insurance the repairing costs equal 0.
Figure 4. Normal form game of the adapted used car game. First element of the payoff vector
corresponds to the sender and the second element corresponds to the receiver with c the amount of repairing costs. Best responses (if determined) are denoted with thick figures. For the game without insurance the repairing costs equal 0.
Sender \ Receiver Pool Sep
High 2-0.25c , 0.75+0.25c 3-0.5c , 0.5+0.5c
Sep 1.75 , 1.38 1.5 , 2
Low 1.25 , 1.88 1 , 1
2.3 Equilibria of the Used Car Games
Signalling games are mostly analysed by using the static concept of weak sequential equilibrium. A weak sequential equilibrium doesn’t only consist of the strategies that are played, but also depend on the beliefs of the players. The strategies that are played and the beliefs of the players combined are often denoted by an assessment.
(1.5-0.5c , -0.5+0.5c) 2 Trade Trade Abstain (Sep) Abstain (Sep) Pool Pool Low price Low price 0.5 1 Nature 0.5 1 High price High price 2 Abstain Pool Trade (Sep) Abstain Pool Trade (Sep) (3 , 3) (2.5 , 2) (2 , 1) (3-c , -2+c) (0 , 1) (1 , 5) (1 , 0.75) (2 , 1) (0 , 1) (1.5 , 3) (2 , 1) (High quality) (Low quality)
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Definition 1. An assessment in an extensive form game is a pair (, ), where
is a profile of behaviour strategies for ;
is a belief function that assigns to every information set a probability measure on the
set of non-terminal nodes in the information set.
A weak sequential equilibrium satisfies the properties sequential rationality and weak consistency. The definitions of these properties followed by the definition of a weak sequential equilibrium are as follows:
Definition 2. An assessment (, ) is sequential rational if for each player i and each
information set Ii of player i,
for each of player i’s behavioral strategies .
Definition 3. An assessment (, ) in a game is weakly consistent if for every information
set I reached with positive probability, (I) is determined using Bayes’ rule and the profile of behavioural strategies .
Definition 4. An assessment (, ) is a weak sequential equilibrium if it satisfies
sequential rationality,
weak consistency.
In a 2x2 signalling game like the original used car game, there can be any or all of the following weak sequential equilibria:
Separating equilibrium, where the sender’s types use different messages.
Pooling equilibrium, where the sender’s types use the same message.
Hybrid equilibrium, where some sender’s types use randomization, others do not.
2.3.1 Weak Sequential Equilibria of the Original Used Car Game
As expected in an opposed interests game, the original used cars game without insurance does not have a separating equilibrium, but this game contains one pooling equilibrium and one hybrid equilibrium. The pooling equilibrium is not a really good outcome of the game, because in this case there is no trade and hence no market. In this hybrid equilibrium there is still no trade of a car, but this is because of the assumption that the receiver prefers no low quality car for a low price. In the hybrid equilibrium there is at least some sharing of information between the players going on, so this is a better equilibrium. When costs increases, the interests of the players become more aligned and the equilibria of the game change and the players start to cooperate more with each other. An overview of the sequential equilibria for different costs is given in Proposition 2.1. These sequential equilibria
9 correspond with the pure and mixed Nash equilibria of the normal form game illustrated in Figure 2.
Proposition 2.1. The weak sequential equilibria in the original used car game are
If 0 ≤ c < 1: A pooling equilibrium, where the sender always asks a high price and the receiver always abstains.
A hybrid equilibrium, where the sender asks a high price when the quality of the car is high and mixes between a low and high price when the quality of the car is low. The receiver always abstains.
If c = 1: A pooling equilibrium, where the sender always asks a high price. The receiver is indifferent between ‘trade’ and ‘abstain’ when he receives a high price and plays ‘abstain’ or a mixed strategy with lower expected payoff if he receives a low price.
If 1 < c < 3: A pooling equilibrium where the sender always asks a high price. The receiver plays ‘trade’ when receiving a high price and plays ‘abstain’ or a mixed strategy with lower expected payoffs if he receives a low price.
If c = 3: A pooling equilibrium where the sender always asks a high price. The receiver plays ‘trade’ when he receives a high price and plays ‘abstain’ when he receives a low price.
A hybrid equilibrium, where the sender asks a high price when the quality of the car is high and mixes between a low and high price when the quality of the car is low. The receiver trades when he receives a high price and abstains when he receives a low price.
A separating equilibrium, where the sender asks a high price when the quality of the car is high and a low price when the quality of the car is low. The receiver trades when he receives a high price and abstains when he receives a low price.
If c > 3: A separating equilibrium, where the sender asks a high price when the quality of the car is high and a low price when the quality of the car is low. The receiver trades when he receives a high price and abstains when he receives a low price.
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2.3.2 Nash Equilibria of the Adapted Used Car Game
For the adapted used car game only a brief summary will be given of the Nash equilibria of the normal form game (see Figure 4), because this game is only used as illustration of the consequences of costly signalling. The strategies in this game are such that the receiver perfectly believes that the sender is telling the truth about the quality of the car or that the receiver believes that the sender is lying. If he believes the sender he plays Sep and if he doesn’t believe the sender he plays Pool.
When costs are low, c < 1, the normal form game (see Figure 4) consists of one Nash equilibrium namely: (High, Pool). This corresponds with the sender always asks a high price and the receiver not believing the signal of the sender. When costs become higher, 1 ≤ c ≤ 3, this Nash equilibrium disappears and a more cooperative equilibrium is achieved. Eventually the equilibrium (Sep, Sep) is reached when c > 3 and this corresponds with a cooperative equilibrium where the sender is telling the truth and the receiver follows the signal.
However, static analyses using Nash and sequential equilibria leave some important questions unanswered. For example: will a population of senders and receivers converge to an equilibrium? If that is the case, how likely is it that such a population of senders and receivers ends up at either the pooling, the hybrid or the separating equilibrium? Therefore it is needed to look at the dynamics of the signalling game and hence a dynamic model is essential to answer these questions.
2.4 Dynamics of the Used Car Games
To find answers on the questions of the previous paragraph, a dynamic model will be used. The model used is the replicator dynamics model, which is the fundamental dynamical model of evolutionary game theory. It describes evolutionary changes in terms of differences between a strategy’s average payoff and overall average payoff of the population. If the difference is positive, the strategy’s share will increase. If the difference is negative, the strategy’s share will decrease. So, this model essentially works as a selection mechanism. The one-population replicator dynamics is the most common replicator dynamics model, but this model can only be used for symmetrical two-player games. However a signalling game consists of two different populations namely: senders and receivers. So, a two-population replicator dynamics model will be used instead, because this model could be used for an asymmetric game. The two-population replicator dynamics captures the basic process of evolution by natural selection in asymmetric games.
The first population of the two-population replicator dynamics model are the senders. In the original used car game the senders have got four pure strategies: (High, High), (High,
Low), (Low, High) and (Low, Low). Let’s denote the proportions of these strategies by x = {x1
11 four pure strategies: (Trade, Trade), (Trade, Abstain), (Abstain, Trade) and (Abstain,
Abstain). The proportions of these strategies are expressed by y = {y1 y2 y3 y4}’. The
proportions add up to 1, so x1 + x2 + x3 + x4 = 1 and y1 + y2 + y3 + y4 = 1, the dynamics live in
the four dimensional space 4 4 where n is the dimensional simplex . The differential equations of the replicator dynamics model are:
Where A is the sender’s 4x4 payoff matrix and B is the receiver’s 4x4 payoff matrix which can be verified from Figure 2.
In the adapted game the senders have three pure strategies: High, Low and Sep. Let’s denote the proportions of these strategies by x = {x1 x2 x3}’. The second population are the
receivers, they have got two pure strategies: Pool and Sep. The proportions of these strategies will be expressed by y = {y1 y2}’. The proportions add up to 1, so x1 + x2 + x3 = 1 and y1 + y2
= 1, the dynamics live in the three dimensional space 3 2 where n is the dimensional simplex . The differential equations of the replicator dynamics model are:
Where C is the sender’s 3x2 payoff matrix and D is the receiver’s 2x3 payoff matrix which can be verified from Figure 4.
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2.5 Dynamical Stability Concepts
In this paragraph the stability concepts that are used to analyse the dynamics of the sequential equilibria of the original used car game, are discussed. The strongest stability property is an evolutionary stable strategy (ESS). If a strategy in a symmetric game is evolutionary stable than this is an asymptotically stable state in the replicator dynamics, because evolutionary stability implies asymptotical stability in the replicator dynamics.
Definition 5. Let be a finite symmetric game and its mixed
extension. Then is an evolutionarily stable strategy (ESS) of if for every mixed strategy (and )
A strategy could also be an neutrally stable strategy (NSS), which is a weaker stability property than ESS. If a strategy in a symmetric game is neutrally stable than this strategy is Lyapunov stable in the replicator dynamics, because neutrally stability implies Lyapunov stability in the replicator dynamics. Lyapunov stability requires solution trajectories that start nearby the state to stay close to all further times (Hutteger and Zollman, 2011).
Definition 6. Let be a finite symmetric game and its mixed
extension. Then is an neutrally stable strategy (NSS) of if for every mixed strategy (and )
However, the stability concepts could only be applied on a symmetric game and not on an asymmetric game. The original used car game is an asymmetric game, so that is why the corresponding symmetrized game (Cressman, 2003) is considered. In the symmetrized game a player is assumed to be the sender or the receiver with equal probability. By taking the expected payoffs, the resulting game is a symmetric game (to which the stability concepts could be applied).
Selten (1980) showed that there is a simple relationship between the strict Nash equilibria of the asymmetric game and the ESSs of the corresponding symmetrized game: a strategy of the symmetrized game is an ESS if, and only if, the corresponding pair of strategies is a strict Nash equilibrium of the asymmetric game. In this thesis, other authors are followed and ESS and strict Nash equilibrium will be used interchangeably when talking about the asymmetric original used car game. Also when I talk about the NSSs of the asymmetric original used car game, I mean the corresponding NSS of the symmetrized game. There is a precise correspondence between these two, just as in the case of ESS (Cressman, 2003).
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Chapter 3
Results
This chapter shows the results of the replicator dynamics used for the models discussed in the previous chapter. First, the dynamical results of the adapted used car game are discussed in order to illustrate the information sharing feature of the players. Thereafter, the stability properties of the sequential equilibria are analysed of the original used car game. Finally, the basins of attractions of the original used car game are discussed.
3.1 Dynamical Consequences of Costly Signalling
The feature of information sharing stays central in the signalling game. In order to illustrate the changes of cooperation in terms of sharing information between the sender and receiver by costly signals, the results of the adapted used car game will be discussed first. In this manner the following analyses of the original used car game becomes more synoptic and clear.
The adapted normal form game without insurance got a strict dominant strategy High and one pure Nash equilibrium (High, Pool). If a game got a dominant strategy the other strategies die out in a replicator dynamics model. This means that x2 and x3 go to zero and
Pool is in that case the best strategy for the receiver, so it is no surprise that the dynamics
converge to the Nash equilibrium independently of the initial state (when costs equal 0). This result means that the players will not cooperate with each other.
When insurance is introduced in terms of repairing costs c, the best responses of the players change and there are five different amounts of costs that are noteworthy to discuss. If the costs are equal to 1, the dynamics converge to a strategy where the sender still always chooses High and the receiver plays a mixed strategy with probabilities that are dependent of the initial state. If the costs are between 1 and 3, the sender chooses High and the receiver
Sep. If costs are equal to 3, the dynamics converge to a strategy where the sender mixes his High strategy with his Sep strategy dependent of the initial state and the receiver plays Sep.
Finally when costs are higher than 3 the dynamics converge to the strict Nash equilibrium (Sep, Sep). The phase portraits with the different costs are shown in Figure 5.
So, the outcomes of a signalling game could be changed by making use of costly signalling (or insurance in this case), such that a non-cooperative game could change into a cooperative game. As can be verified in Figure 5, for higher costs the initial points converge
14 to a more cooperative equilibrium, which makes sense, but is this always the case? In other words, would a better equilibrium be reached, in terms of the amount of information sharing between the players, if the costs of the signal are increased?
Figure 5. Phase portraits of the adapted used car game for different costs. Black and grey dots
indicate stable and unstable rest points respectively. The black dots of the figures where costs equal 1 and costs equal 3 could lie on the whole line, dependent of the initial state.
3.2 Dynamical Stability of the Sequential Equilibria
Having analysed the adapted game in previous paragraph, the original used car game will be considered now. In this paragraph the stability properties of the sequential equilibria of Proposition 2.1 are analysed with the stability concepts that are discussed in the previous chapter. This is a reasonable thing to do before looking at the basin of attractions. As already mentioned in the previous chapter, the stability concepts are applied on the corresponding symmetrized original used car game. Once again, if a sequential equilibrium is called an ESS or an NSS, then the corresponding NSS or ESS in the symmetrised game is meant.
As mentioned before, if the normal form of the used car game (for some c) consists of a strict Nash equilibrium then it is an ESS in the symmetrized game, which is an asymptotically stable state in the replicator dynamics. There is a strict Nash equilibrium in the original used car game when c is higher than 3, which corresponds with the separating equilibrium. Hence, the separating equilibrium is an ESS and it is the only equilibrium for that
High, Pool High, Sep Low, Sep
Low, Pool
Sep, Pool Sep, Sep c = 3
High, Sep Low, Sep
High, Pool Low, Pool
Sep, Pool Sep, Sep 1 < c < 3
Low, Pool
High, Pool High, Sep Low, Sep
Low, Sep Sep, Pool Sep, Sep 0 ≤ c < 1
Low, Pool
High, Pool High, Sep c = 1
Low, Sep Sep, Pool Sep, Sep
Low, Sep
High, Pool High, Sep Low, Pool
Sep, Pool Sep, Sep c > 3
15 amount of costs. For this reason the replicator dynamics will probably converge to this strategy in this case. However, the separating equilibrium is not a strict equilibrium when c is equal to 3, but it can be shown that in this case the separating equilibrium is an NSS, as can be verified in Proposition 3.1.
Proposition 3.1. The separating equilibrium of the original used car game is neutrally stable
for c = 3 and evolutionarily stable for c > 3 in the corresponding symmetrized version of the original used car game.
Proof. See appendix B.
Pooling equilibria are elements of a set of equilibrium, so that is why the pooling equilibria are not strict. This is easy to see in the normal form representation of the used car game (Figure 2), because when the receiver changes his strategy in the off-equilibrium path the payoffs don’t change. For this reason the pooling equilibrium will never be an ESS, so not evolutionary stable. However, this equilibrium turns out to be neutrally stable for particular c as can be verified in Proposition 3.2. As mentioned in the previous chapter, neutral stability implies Lyapunov stability in the replicator dynamics, so the dynamics may converge to the pooling equilibrium for some c.
Proposition 3.2. The pooling equilibrium of the original used car game is unstable for 0 ≤ c
< 1 and c = 3, and neutrally stable for 1 ≤ c < 3 in the corresponding symmetrized version of the original used car game.
Proof. See appendix C.
The hybrid equilibrium is also not a strict equilibrium since it is a mixture of strategies and a mixture can never be strict. Nonetheless this equilibrium is also neutrally stable for particular c in the symmetrized game as is shown in Proposition 3.3.
Proposition 3.3. The hybrid equilibrium of the original used car game is neutrally stable for
0 ≤ c < 1 and neutrally stable for c = 3 in the corresponding symmetrized version of the original used car game.
Proof. See appendix D.
Hence, the only asymptotically stable strategy is the separating equilibrium, but this doesn’t mean that the pooling and hybrid equilibria are unimportant. As is shown in the dynamical analyses of the Sir Philip Sidney game, the dynamics converge sometimes to a hybrid equilibrium even when an ESS separating equilibrium exists (Hutteger and Zollman, 2010). In this original used car game there are situations (0 ≤ c < 3) where there doesn’t exist a separating equilibrium. So, where do the dynamics converge to for these costs?
16
3.3 Basins of Attractions of the Original Used Car Game
Now that the stability properties of the sequential equilibria are analysed, the question of previous paragraph could be answered by looking at the modelled basins of attractions. The basin of attraction is the set of initial points leading to long time behaviour that approaches a particular attractor. Interest lies in how the dynamics changes from the game without insurance to the game with insurance and how much percentage will converge to which equilibrium. It is expected from the results of the adapted game that the players start to cooperate more with each other when c increases. The dynamical results are analysed by increasing the costs from 0 to 3.2, taking steps of one tenth. The basins of attractions of the dynamics are modelled by looking at the convergence of 100 random initial points for each game with different costs. The results are analysed for five different cost categories in which the sequential equilibria changes. An important thing to notice is that there are more Nash equilibria than sequential equilibria in the normal form game, so the initial points might not always converge to a sequential equilibrium. To make the thesis more readable let’s simplify the notation of possible pure strategies, as is illustrated in Table 1.
Table 1. Labels of all possible pure actions of both players. The first four are the sender’s
strategies and the last four are the receiver’s strategies.
Label Description
HH Always signal High
HL Signal High when High quality and Low when Low quality LH Signal Low when High quality and High when Low quality LL Always signal Low
TT Always Trade
TA Trade when receiving High signal and Abstain when receiving Low signal AT Abstain when receiving High signal and Trade when receiving Low signal
AA Always Abstain
3.3.1 Repairing Costs below 1
The original used car game without insurance (repairing costs equal to 0) and the original used car game with repairing costs lower than 1 have got two sequential equilibria, namely: one pooling and one hybrid equilibrium as can be verified from Proposition 2.1. From the previous paragraph it is known that the hybrid equilibrium is the only neutrally stable equilibrium for this amount of costs. Furthermore, the initial points converge most of the time to the hybrid equilibrium as can be verified from Figure 7. But when costs increases, the dynamics start to converge less to the hybrid equilibrium and some other stable points will be reached, also the pooling equilibrium. However in the other stable points, where the receiver plays a mixed strategy, the receiver still plays ‘abstain’ for almost 100%. In the hybrid equilibrium the sender chooses a high price when the quality of the car is high and mixes between a high price and a low price when quality of the car is low. This is actually the better
17 equilibrium of the two possible sequential equilibria in terms of information sharing. This is quite a remarkable result because after all, the hybrid equilibrium is only a mixed strategy and the pooling a pure strategy. The convergence is reached after a period of unclear switching of strategies which make sense because the hybrid equilibrium is only neutrally stable, as can be verified in Figure 8.
Figure 7. Percentage of convergence of 100 initial points to particular strategies of the sender
and the receiver for costs below 1. ‘Pooling’ and ‘Hybrid’ in brackets mean pooling equilibrium respectively hybrid equilibrium.
Figure 8. First 500 steps of the replicator dynamics with initial point (0.1515, 0.3593, 0.2511,
0.2381, 0.3370, 0.1074, 0.2778, 0.2778) without insurance converge to (0.8926, 0.1074, 0, 0, 0, 0, 0, 1). 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Per ce n tages Costs
mix HH,HL; mix TT,AA mix HH,HL; mix TT,AT,AA mix HH,HL; mix TA,AA mix HH,HL,LH,LL; AA mix HH,HL,LH; AA HH; mix AT,AA mix HH,HL; mix AT,AA HH; AA (Pooling) mix HH,HL; AA (Hybrid)
18
3.3.2 Repairing Costs of 1
There is only one sequential equilibrium in the game when the repairing costs equal 1, this is the pooling equilibrium. Only for these costs, the pooling equilibrium changes a bit as is shown in Proposition 2.1. Now, the receiver plays any strategy when he receives a high price and he plays ‘abstain’ or he plays a mixed strategy with lower expected payoff when he receives a low price. The initial points converge 100% to HH for the sender and a few mixes of strategies for the receiver as is shown in Figure 9. Hence, it looks like the dynamics converge to different strategies, but they all correspond to the same pooling equilibrium. While the hybrid equilibrium for c < 1 was not a fully cooperative equilibrium, there was at least some sharing of information going on between the players. Now, the dynamics converge to a totally non-cooperative equilibrium, hence the amount of sharing information actually decreases by costly signalling in this game. This is an important result, because from the analysed adapted used car game and previous research, one would expect that the amount of sharing information should increase when costs increases.
The reason why the amount of information is actually decreasing by costly signalling in this particular game is because the receiver almost always plays ‘abstain’ for c < 1. So, it doesn’t matter if the sender always asks a high price or if the sender asks a high price for a high quality car and a low price for a low quality car, the sender will always get a payoff of 1. Furthermore, the basin of attractions are modelled that start with a random initial point, so it is more likely to end up at the hybrid than the pooling equilibrium for c < 1 (, because the payoffs are equal). When c becomes 1, the receiver plays sometimes ‘trade’, hence then it becomes more attractive to ask always a high price for the sender and the amount of sharing information decreases.
3.3.3 Repairing Costs between 1 and 3
For these repairing costs the only sequential equilibrium is once again the pooling equilibrium. As can be verified from Figure 9, the dynamics almost always converge to this sequential equilibrium for these repairing costs category. Only when costs becomes close to 3 the dynamics converge for a small percentage to a cooperative equilibrium. Notice that, when costs become higher, ‘abstain’ is played in the off-equilibrium path of the pooling equilibrium. Otherwise, the type 2 sender gets an incentive to deviate. These outcomes are in line with the stability properties of the pooling equilibrium of Proposition 3.2. Like in previous costs category the players don’t share information at all. The path to convergence for costs equal to 2 is shown in Figure 10, which looks smoother than the convergence to the hybrid equilibrium in Figure 8. Also the convergence is reached after around 175 steps instead of 400. This is probably because the pooling equilibrium is the only sequential equilibrium in this case.
19
Figure 9. Percentage of convergence to strategies for 100 initial points for costs in the range
of 1 till 2.9. ‘Pooling’ in brackets stands for the pooling equilibrium.
Figure 10. First 300 steps of the replicator dynamics with initial point (0.3178, 0.1186,
0.2881, 0.2754, 0.0971, 0.0686, 0.2857, 0.5486) with cost equal to 2 converge to (1, 0, 0, 0, 0.0058, 0.9942, 0, 0).
3.3.4 Repairing Costs of 3
When costs of repairing equals 3 there are three sequential equilibria in the game namely: a pooling equilibrium, a hybrid equilibrium and a separating equilibrium. Only two of them are 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 Per ce n tages Costs HL; TA HH; TA (Pooling) HH; mix TT,TA,AA(Pooling) HH; mix TT,TA,AT (Pooling) HH; mix TT,TA,AT,AA (Pooling) HH; mix TT,TA (Pooling)
20 neutrally stable strategies as can be verified from Proposition 3.1-3.3, which are the separating and the hybrid equilibrium. The initial points converge to the hybrid or separating equilibrium as can be seen in Figure 11, which is in line with the stability properties. Hence the initial points converge 50% of the times to the separating equilibrium, 50% to the hybrid equilibrium and 0% to the pooling equilibrium. This means that the players start to cooperate with each other again. Again, the sender doesn’t always ask for a high price anymore and the receiver perfectly believes what the sender is indicating by trading when he receives a high price and abstaining when he receives a low price, which are in agreement with the preferences of the players.
3.3.5 Repairing Costs of Higher than 3
If the repairing costs are higher than 3, there is only one equilibrium namely: a separating equilibrium. So it is no surprise, especially because the separating equilibrium is an ESS in this case, that all the initial points converge to this sequential equilibrium which also can be verified from Figure 11. In this equilibrium the players perfectly cooperate with each other by honest signalling. For all initial points the replicator dynamics converge fast compared to the other costs categories, an example is given in Figure 12.
Figure 11. Percentage of convergence of 100 initial points to particular strategies of the
sender and the receiver for costs in the range of 3 till 3.6. ‘Separating’ and ‘Hybrid’ in brackets mean separating equilibrium respectively hybrid equilibrium.
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 3 3,1 3,2 3,3 3,4 3,5 3,6 Per ce n tages Costs HL; TA (Separating) mix HH,HL; TA (Hybrid)
21
Figure 12. First 150 steps of the replicator dynamics with initial point (0.1789, 0.3053,
0.1211, 0.3947, 0.1102, 0.2161, 0.2966, 0.3771) with cost equal to 3.5 converge to (0, 1, 0, 0, 0, 1, 0, 0).
3.3.6 Expected Payoffs of the Players
In the previous five subchapters the point of interest is the amount of shared information between the players, which is the most important concept of this thesis. However, the corresponding expected payoffs of the strategies might also give some important insights into the analyses of this particular game. In Figure 13 the expected payoffs of the converged strategies of Figure 7, 9 and 11 are shown, where the costs increases from 0 to 3.6 once again.
Figure 13. 2 The expected payoffs of the converged strategies, where costs increases from 0 to
3.6.
2
For costs below 1, the payoffs of the sender and receiver look both equal to 1. However, the sender’s payoffs are slightly above 1 and the receiver’s payoffs are slightly below 1 for these costs.
0 0,5 1 1,5 2 2,5 3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3 3,1 3,2 3,3 3,4 3,5 3,6 Exp e cte d Pay o ff s Costs Sender Receiver
22 The expected payoffs in Figure 13 confirm the thoughts about the decrease of information sharing between the players. For c < 1, the receiver almost always abstains so the payoff of both players will always be close to 1. When 1 ≤ c < 3, the sender gets an incentive to lie about the quality of the car by always asking a high price, because the receiver is best off by trading for a high price. Finally when c 3, the sender is best of by perfectly telling the truth and the receiver perfectly believing the sender. So in terms of expected payoffs, the receiver gets his highest payoff when c 3 and the sender gets his highest expected payoff for c = 1. Hence, once again, the strategy of the sender doesn’t matter when the quality of the car is low for c < 1, because the receiver always abstains and they both get a payoff of 1. This causes the decrease in information sharing between the players.
23
Chapter 4
Discussion
This thesis shows that costly signalling doesn’t always improve the dynamical outcome of the game in terms of the amount of information sharing between the players in an adapted game of Akerlof (1970). By focusing on the stability properties of the sequential equilibria of the game there seems to be no rare results. The pooling and hybrid equilibria of the game are sometimes Lyapunov stable and the separating equilibrium is sometimes Lyapunov stable and sometimes asymptotically stable in the replicator dynamics. But by looking at the basins of attractions, the hybrid equilibrium turns out to have larger basins of attractions than the pooling equilibrium for low costs, which is the reason of the decrease in the amount of shared information.
Although there could be said that this decrease of shared information in this game isn’t really relevant, because the receiver always plays ‘abstain’ in this hybrid equilibrium, hence there is no market anyway. Still, it is important, because in this hybrid equilibrium there are some senders who are telling the truth about the quality of the car, whereas in the pooling equilibrium there are no senders that are telling the truth about the quality of the car whatsoever. Hence, while there is no detriment in the outcome (in terms of payoffs) of the game, there is a detriment in the communication feature of the game.
Traditionally, research that used experiments by making use of signalling games compared two hypotheses namely:
But, these hypotheses don’t distinguish between the amounts of communication, so they don’t distinguish the separating equilibrium from the hybrid equilibrium. Since the last 10 years, there is an upcoming philosophy that the hybrid equilibrium plays an important role in the explanation of information transfer that has been underappreciated in the past. For example, Wagner (2013) used the replicator dynamics and the best response dynamics on the well-known Spence’s model of education. He found, among other things, that the partially communicative mixed (hybrid) equilibria are quite important dynamically in that game.
This thesis is in line with this philosophy of the underappreciation of the hybrid equilibrium, because the hybrid equilibrium ensures a decrease in information sharing between the players when costly signalling should stimulate the opposite. However, it might be said that in this thesis, costly signalling is used in a slightly different manner. In most papers the actual signal is giving some costs, whereas in this thesis the payoffs of the eventual
24 outcome is changed to give the same effect. It is hard to say if this decrease in information sharing could be a general result that could occur for other signalling games as well. This is because, it seems that every signalling game reacts differently, in terms of dynamics, on costly signalling.
Further research could explore other signalling games by evolutionary dynamics as well or extend this game with more strategies or more states of the world. Another interesting research would be, if other dynamics models show different results than the evolutionary dynamics model. Furthermore, it would also be interesting to explore the relevance of the hybrid equilibrium in real life.
This thesis suggests that policy makers or agencies that could influence a market to stimulate cooperation have to be careful, because the results might not always be satisfying in terms of cooperation between the players.
25
Chapter 5
Conclusion
This thesis tries to answer the following question: how and why do the dynamics change for different costly signals in this adapted game of Akerlof (1970)? The other more general question that is tried to be answered is: could increasing costs of a signal only increase the amount of shared information, or could it also decrease the amount of shared information? In order to answer these questions, there are two signalling games analysed, the original used car game and an adapted used car game. The interests of the players in both used car signalling games are changed from opposed to aligned by a cost variable. This costs variable is increased from 0 to 3.6, in steps of 0.1. The adapted used car game is used to make the costly signalling consequences in terms of information sharing between the players more clearly by using a two-population replicator dynamic model. The original used car game is analysed by using stability concepts and a two-population replicator dynamics model. The stability concepts are used to analyse the stability properties of the sequential equilibria. The replicator dynamics are used to model the basins of attractions by looking at the convergence of a set of initial points.
To illustrate the consequences of costly signalling, the original used car game in the thesis is adapted such that the strategies of the receiver are: perfectly believing the sender and don’t believe the sender at all. It turns out that by increasing the costs variable the amount of sharing information is monotone increasing. So when there are no costs in the adapted used car game, all the random initial points converge to the non-cooperative equilibrium. If costs increase the dynamics start to converge more and more to a cooperative equilibrium. If costs are high enough, the players cooperate fully with each other.
To analyse the stability properties of the sequential equilibria in the original used car game, the game needed to be symmetrized. Only in the symmetrized game concepts such as Evolutionary Stable Strategies (ESS) and Neutrally Stable Strategies (NSS) are applicable. The separating equilibrium is the only equilibrium that is an ESS most of the time, when it exists. However the separating equilibrium could also be an NSS for a particular amount of costs. The pooling equilibrium is sometimes an NSS and sometimes unstable, when it exists. The hybrid equilibrium, which is a mixed equilibrium, is always an NSS when it exists. Evolutionary stability and neutrally stability implies asymptotical stability and Lyapunov stability respectively in the replicator dynamics.
To investigate the dynamical results, the basins of attractions of the dynamics model are modelled by looking at the convergence of 100 random initial points for each game with different costs. The changes in convergence results are in line with the stability properties of
26 the sequential equilibria. When there is only one sequential equilibrium that is an ESS, the dynamics converge to this particular equilibrium for 100%. When there is one sequential equilibrium that is an NSS, the dynamics converge most of the time to this equilibrium. When there are two NSS, the dynamics converge for approximately the same percentage to each of these equilibria. These results hold of course only for this particular game.
In the original used car game the amount of information sharing between the players is not always increasing. For 0 ≤ c < 1, the initial points converge most of the time to the hybrid equilibrium, while there exists also a pooling equilibrium. When 1 ≤ c < 3 the initial points converge most of the time to the pooling equilibrium. So the amount of information sharing between the players decrease, because in the hybrid equilibrium the players share more information than in the pooling equilibrium. If c is equal to 3 the initial points converge for 50% to the hybrid equilibrium and 50% to the separating equilibrium, so the amount of sharing information between the players is increasing again. When c is higher than 3, the initial points converge for 100% to the separating equilibrium, hence the players fully cooperate with each other.
To answer the first research question, the convergence to the equilibrium depends mostly on the stability properties of the sequential equilibria and the amount of stable equilibria in the game. Also the strength of the stability properties of the sequential equilibria is important. The second question is a more general question, so it is hard to say anything in general by only the results of this particular game. But in this game it is the case that when the costs increases, convergence of the dynamics changed from hybrid equilibrium to pooling equilibrium. This means that the amount of information sharing between the players actually decreases. Hence, costly signalling can actually aggravate the game in terms of information sharing between the players.
27
Chapter 6
References
David K. Lewis. 1969. “Convention: A Philosophical Study”.
Michael Spence. 1973. “Job Market Signaling”. The Quarterly Journal of Economics, vol. 87, No. 3, pp. 355-374.
Maynard J. Smith. 1991. “Honest signaling: The Philip Sidney game”. Animal Behaviour, vol. 42, pp. 1034-1035.
Simon M. Huttegger, Brian Skyrms, Rory Smead and Keven J. S. Zollman. 2010.
“Evolutionary dynamics of Lewis signaling games: signaling systems vs. partial pooling”.
Synthese.
Elliot O. Wagner. 2012. “Deterministic chaos and the evolution of meaning”. The British Journal of Philosophy of Science, vol. 63, No. 3, pp. 547-575.
George A. Akerlof. 1970. “The Market for “Lemons”: Quality Uncertainty and the Market
Mechanism”. The Quaterly Journal of Economics, vol. 84, No 3, pp. 488-500.
Simon M. Hutteger and Kevin J. S. Zollman. 2010.“Dynamic stability and basins of
attraction in the Sir Philip Sidney game”. Proceedings of the Royal Society B: Biological
Sciences, vol. 277, pp. 1915-1922.
Ross Cressman. 2003. “Evolutionary Dynamics and Extensive Form Games”. Cambridge, MA: MIT Press.
Reinhard Selten. 1980. “A note on evolutionary stable strategies in asymmetrical animal
conflicts”. Journal of theoretical Biology, vol. 84, pp. 93-101.
Simon M. Hutteger and Kevin J. S. Zollman. 2011. “Evolution, dynamics and rationality: the
limits of ESS methodology”. Evolution and Rationality: Decisions, Co-operation and Strategic
Behaviour. Pp. 63-83.
28
Chapter 7
Appendices
7.1 Appendix A
Sequential equilibria of the original used car game
The steps that are followed to search for weak sequential equilibria, which have to satisfy sequential rationality and weak consistency, in the original used car game are as follows:
1. Decide which equilibrium (separating, pooling or hybrid) is tried to be found.
2. Assign a strategy (a signal for each type) to Player 1 (the sender), because in total there are 2 separating, 2 pooling and 4 hybrid equilibria possible in a 2x2 game.
3. Derive the beliefs for Player 2 (the receiver) according to Bayes’ rule at each information set reached with positive probability along the equilibrium path. When information sets are never reached along the equilibrium path (which occurs with a pooling equilibrium), use arbitrary beliefs for these information sets. This is done to satisfy the weak consistency property.
4. Determine Player 2’s best responses.
5. Check whether Player 1 has an incentive (in view of Player 2’s best responses) to deviate from the strategy that is assigned in step 2 (for both states of the world). If both types of Player 1 don’t want to deviate, the strategy is a weak sequential equilibrium. If at least one type of Player 1 wants to deviate, the strategy isn’t a weak sequential equilibrium.
All possible weak sequential equilibria (that are found by following the steps) are illustrated below. The other separating, pooling and hybrid equilibria which are not mentioned, don’t exists for any c. Is the strategy played, is the payoff variable and is the state of the game, where corresponds with a high quality of the car and with a low quality of the car. The beliefs of player 2 are updated with Bayes’ rule denoted with , where is the arbitrarily off equilibrium belief which cannot be updated by Bayes’ rule. The strategies of Player 1 are
29
Separating equilibrium (, where the sender’s types use different messages)
Player 1 strategy
Beliefs player 2
Best responses player 2
Against L Against H
Is this a weak sequential equilibrium?
Sender type 1 gets a payoff of 3 along the equilibrium path, because Player 2 trades for a high price. If he deviates he gets 2, because Player 2 abstains for a low price. So, he will not deviate.
Sender type 2 gets a payoff of 0 along the equilibrium path, because Player 2 abstains for a low price. If he deviates he gets 3 - c, because Player 2 trades for a high price. So, he will deviate if c < 3.
So, this separating equilibrium exists if c 3.
Pooling equilibrium (, where the sender’s types use the same message)
Player 1 strategy
30
Beliefs player 2
Player 1 plays always a high price, so the right side of the extensive form game is never reached. This means that arbitrary beliefs are needed for the information nodes on the right side, because they cannot be updated by Bayes’ rule.
Best responses player 2
Against L Against H
Is this a weak sequential equilibrium?
Sender type 1 gets a payoff of at least 2 along the equilibrium path. If he deviates he gets 2 or less, so he will not deviate.
For sender type 2 it is more complicated, because his decisions depend on the costs and the best responses of Player 2, which depend on the arbitrary beliefs of Player 2.
31 So, this pooling equilibrium exists if 0 ≤ c ≤ 3.
Hybrid equilibrium (, where some sender’s types use randomization, others do
not)
Player 1 strategy
Beliefs player 2
Best responses player 2
Against L Against H
Is this a weak sequential equilibrium?
Sender type 1 gets a payoff of at least 2 along the equilibrium path. If he deviates he gets 2, because Player 2 abstains for a low price. So, he will not deviate.
For Sender type 2 the payoffs by choosing a high price need to be the same as choosing a low price, because otherwise he will deviate to one side instead of mixing.
