The survival of ambiguous strategies in Evolutionary Signalling Games Fije van Overeem 10373535 Bachelor thesis Credits: 18 EC Bachelor Opleiding Kunstmatige Intelligentie University of Amsterdam Faculty of Science Science Park 904 1098 XH Amsterdam Supervisor Thomas Brochhagen Institute for Language and Logic Faculty of Science University of Amsterdam Science Park 107 1098 XG Amsterdam June 26th, 2015 1
Abstract It is argued that one of the reasons for ambiguity to permeate natural language is that ambiguous language is less complex to use than precise language. Aside from that, during communication there is context available which often helps to disambiguate. Lewisian signalling games can model ambiguous and precise language. By means of evolutionary signalling games the stability of ambiguous strategy users is tested with different costs for a precise strategy. It is found that under certain conditions ambiguous strategies are favourable, although ambiguous players will always benefit from playing with precise players. The initial ratio ambiguous:precise players in the population does not influence the results significantly. An interpretation of the results is that natural language knows ambiguity because there is equivalent precise language available. For future research it is proposed to also make context available to signalling game players. 2
Contents Introduction 4 Lewisian Signalling Games........................ 5 Evolutionary Signalling Games...................... 7 Theoretical foundation.......................... 7 Assumptions................................ 8 Methods 9 Setting up the signalling games...................... 9 Choosing the variables.......................... 10 Population ratios and χ 2 -tests of homogeneity........... 10 β-payoff for precise senders.................... 13 Probability of mutation....................... 13 Results 15 Varying the β-value............................ 15 Varying the probability of mutation.................... 17 Conclusion 19 Discussion 20 Appendices 22 Appendix A: Classes Ambiguous Player & Precise Player........ 23 Appendix B: Class Evolutionary Signalling Games........... 25 Appendix C: Code for experiments.................... 26 Appendix D: Plots............................. 34 Appendix E: χ 2 -test for homogeneity of populations.......... 38 3
Acknowledgments I wish to express my sincere thanks to Thomas Brochhagen for his ideas, comments and discussion. Introduction Over the last few decades research has been conducted about where and why natural languages are structured the way they are [7, 2]. For this goal it is interesting to investigate how natural language differs from formal language. Formal languages are required to be concise, literal and unambiguous, whereas natural languages are often redundant, figurative and imprecise. Formal language was designed with these properties in mind. Natural language, in contrast, has developed over years without direct human supervision and has become so complex to grasp that we are not fully aware of all its underlying structures. Mainly because of this fundamental difference between formal and natural languages it is a difficult task to program machines to derive meaning from natural language. To achieve a complete model of natural language understanding, it is necessary to investigate why semantic features have emerged in the first place and how they remain existent. It is to achieve this goal that linguists often search for semantic features that are universal: semantic features that are not only part of all existing natural languages, but are also frequently used by speakers of the language itself. The semantic features that only occur in few natural languages or that are not widely used in communication, are not of major interest to linguists because conceivably, the semantic features that are universal are well-adapted for language use and could thus provide information about why these parts of language are stable. It would have a significant impact on our understanding of the underlying structures of semantic features as a whole if we were to formalise the exact semantic features that are widely spread and occur in every language. One of these widespread semantic features is ambiguity. Ambiguous aspects can be found in every natural language and come in many forms. This is remarkable, because it seems suboptimal that language users choose to express themselves with ambiguous sentences. In many cases there are more precise, unambiguous equivalent sentences available to express themselves. Examining the sentence Alice saw Bob standing on a hill, one can conclude that there are at least two meanings of this sentence that can be derived from its syntactical structure: one where Alice is standing on a hill, and one where Bob is. This type of ambiguous sentences among other types permeate languages even though they can interfere with successful communication whereas a more precise sentence e.g., Alice saw Bob, while she was standing on a hill, would make the meaning of the sentence more 4
clear. However, in some cases ambiguity can be useful. For example the case of partial conflict where the speaker benefits from a misconception of the meaning of a sentence [8]. One could imagine politicians that make imprecise promises that are open for interpretation. They would do this in order for the promises to stay plausible, they do not want to make promises they obviously can t keep. At the same time, the politicians still want to come across as if they re making larger promises than they actually do. In such a case there is a difference between the interest of the sender and the receiver of the information. But partial conflict cannot be the only reason for the existence of ambiguity, as ambiguous language is also used when interlocutors, both the sender and the receiver, share a common interest. It is assumed that in these cases ambiguity can be explained by the fact that ambiguous language is often less complex to use than precise language[12], and misconception can be prevented because all parties involved in communication have access to a context in which communication occurs [4]. The aim of this research is to investigate under what circumstances ambiguous language use is well-suited for communication in cooperative settings and it will do so by exploiting the features of evolutionary signalling games (ESG). The topic of ESG s will be discussed later on; first will be explained what signalling games are: Signalling games are used as a tool to model communication between individuals by means of game theory and were first introduced by Lewis in Convention [6]. Lewisian Signalling Games A Lewisian Signalling Game (LSG) is a simple kind of signalling game that models communication between two interlocutors. The game consists of the following: A sender, S A receiver, R A set of events, E A set of actions, A = E A set of signals, S In the game, an event e E gets arbitrarily allocated to the receiver. This is the information that the sender should communicate to the receiver, via a signal s S. After the signal is revealed to the receiver, the receiver picks an action a A. If 5
t 1 m 1 a 1 t 1 m 1 a 1 t 2 m 2 a 2 t 2 m 2 a 2 Figure 1: Two interchangeable signalling systems the event is the same as the action, (e = a), the game is successful and both players receive a payoff of one. If not, the game has failed and the payoff for both players is zero. This utility function (U) can be defined as follows: { 1 if e = a U S,R = 0 otherwise Notice that at first, if no convention is in place, the signals have no meaning. All events have an equal probability of allocation to the sender. The probability of successful signalling thus depends on the amount of events. For example, if there would be two events (and thus two actions) and two signals, the probability of success would be 1 2. By use of LSG s it can be demonstrated how signals can become meaningful. Primarily, the receiver has no reason to choose any particular action from the set, as the signal that the sender chose was completely random. After a certain amount of signalling between the players, a convention can arise as to which signal has which meaning. This is called a signalling system. These systems do not necessarily emerge: when a player s eventual strategy of choosing an event or action does not depend on any signal, there is no case of a signalling system. It depends on the amount of signals with respect to the amount of events whether a signalling system arises. If the amount of events and signals are the same, it is demonstrated that there will always arise a signalling system. Optimal signalling systems arise when there are as many signals as events and there is thus a one-to-one mapping between them. See figure 1 for two equiprobable one-to-one mapping signalling systems with two events and two signals [9]. These signalling games can be extended, for example when there are multiple signallers who play amongst each other and thus have to create convention between more of them. Some players could have a smaller dictionary than other players which would make them use ambiguous signals, signals that have more than one meaning. Conversely, when two signals match with the same event we can speak of a synonymous strategy (see figure 2). 6
t 1 m 1 a 1 t 1 m 1 a 1 t 2 m 2 a 2 t 2 m 2 a 2 Figure 2: Synonymous and ambiguous signalling systems Evolutionary Signalling Games An ESG is a type of signalling game that relies on the assumption that natural languages are self-replicating systems. When a self-replicating system is subject to variation and selection, it is possible that evolutionary processes can be distinguished [3]. These processes are characterised by mutation of player strategies or by players that inherit a new strategy from other (often more successful ) players, but in this research there is chosen for mutation. Intuitively, a player should be more resistant against mutation of its strategy when the strategy provided the player with a high payoff with respect to players with other strategies. It is interesting to examine how the ratio of these strategies changes in a population during mutation. Depending on the circumstances that the strategies are subject to, after a number of iterations the strategy-ratio can stabilise. If the percentage of players of a certain strategy type does not change anymore after iteration, the strategy is likely to be evolutionarily stable[10, 5]. Theoretical foundation Research about the emergence and evolutionary fitness of semantic features that aims to model communication by means of game theory often exploit Lewis s idea of using signalling games[1, 8, 4]. As mentioned earlier, Lewis first introduced the framework of signalling games in 1969 in his book Convention. By using Lewisian signalling games it can be demonstrated that two interlocutors can assign meaning to previously meaningless signals when the interlocutors share a common interest [8]. This type of signalling can model the emergence of meaning to a language. However, Lewisian signalling games rely on the assumption that an optimal signalling system is characterised by a bijection between signals and events, which results in a one-to-one mapping between signals and events. This assumption is supported by Trapa and Nowak who demonstrated, using evolutionary signalling games, that it is indeed only the one-to-one states that are evolutionarily stable [11]. Ambiguous signals cannot be represented by a one-to-one mapping and intuitively they should be evolutionarily stable as ambiguity in natural languages is universal 7
[3]. Therefore, Jäger has investigated under what circumstances the assumption of Lewisian signalling games can be rejected [3]. Jäger demonstrated that optimal signalling systems are not necessarily representative of natural languages [3]. This means that ambiguous signalling, a suboptimal strategy - because a one-to-many mapping does not guarantee successful signalling - can become evolutionarily stable states as well. At least two explanations are argued for by Santana as to why one-to-many mapping can be evolutionarily stable [8]. The first one is that precise, one-to-one signalling, is in fact costly because these signals are more complex to use for a sender. Secondly, Santana argues, ambiguous strategies can be successful because in the real world there is context available during communication, which is why he argues that receivers in a signalling game should have access to context as well. Santana researched the stability of ambiguous strategies in evolutionary signalling games with both costly signalling and context as extra features, from which he concluded that under these circumstances, ambiguous strategies can indeed become stable states [8]. However, Santana used cost in a different way then will be done in this thesis. Degen et al. have conducted similar research with only costly signalling as an extra feature and concluded that in evolutionary signalling games when an unambiguous alternative is available, communication is more likely to succeed. Furthermore, they demonstrated that there was more failure of communication as the cost of unambiguous alternatives increased [2]. It is not yet known at exactly what cost costly signalling makes ambiguous strategies evolutionarily stable. This research aims to investigate under which circumstances ambiguous strategies can survive in evolutionary signalling games when there s a cost to precise signalling. Assumptions We assume that an ambiguous strategy can be representative for ambiguous natural language. It is conventional to represent ambiguous language as is done in this research. However, there is a choice to be made if we look at different ambiguous strategies. First of all, here we limited the number of meanings assigned to one signal to two, but it could just as well be three or four meanings. This is not only done for simplicity reasons, but it is also to compensate for the fact that we did not make any sort of context available to the players, which would have been in favour of ambiguous signalling; three or more possible meanings for an ambiguous signal would have been less likely to exist in the real world, because context can eliminate a few optional meanings during communication. However, the number of two meanings is not the only righteous one, and in future research these ambiguous signalling systems should thus not be eliminated per se. 8
a s 1 a a s 1 a b s 2 b b s 2 b Figure 3: Two different ambiguous strategies Methods Setting up the signalling games Consider a population of players that either have a precise or an ambiguous strategy. For the chosen representation of an ambiguous strategy, see the right signalling system in figure 3. If two players get randomly assigned to play a signalling game against each other, there are four possible games. Which player is the sender and which player is the receiver, is also determined randomly. In a population where the ratio ambiguous:precise players is 50:50, the probability of all four scenario s is 0.25. These are the possible scenarios: A sender with an ambiguous strategy: S A A sender with a precise strategy: S P A receiver with an ambiguous strategy: R A A receiver with a precise strategy: R P If we let two precise players play against each other we get a Game(S P,R P ) of which we know that it will succeed. As mentioned above, in LSG s there is a payoff of either 1 (success) or 0 (failure). However, we will assign a cost to precise sending. As mentioned earlier, the motivation to do this is because precise communication is more complex to use, which natural language users seem to elude. If we assign a certain cost c to sending precise signals, the payoff after success will be 1 c. We will call this number β. The possible actions that an ambiguous receiver can assign to a meaning are equiprobable to be chosen by the receiver. This makes the mean payoff for the four different situations as follows: Game(S P,R P ): U = β Game(S P,R A ): U = 1 2 1 2 β + 1 2 1 = 1 4 β + 1 2 9
Game(S A,R P ): U = 1 2 1 Game(S A,R A ): U = 1 2 1 2 + 1 2 1 2 = 1 2 If we know the initial population ratio of ambiguous:precise players, we can calculate the expected utility for both ambiguous and precise players. If we take a starting ratio of 50:50 we expect that all four situations occur the same amount of times. For the average ambiguous player the payoff 1 will then be: Ambiguous: 1 4 β+ 1 2 + 1 2 + 1 2 + 1 2 4 = 2+ 1 4 β 4 = 1 2 + 1 16 β Precise: β+β+ 1 4 β+ 1 2 + 1 2 4 = 9 16 β + 1 4 What if we do not make precise signalling costly? β would just be 1. The mean payoff for the ambiguous players would be 9 16 and the mean payoff for precise players would be 13 16, a lot higher. This means that, depending on the probability that a player s strategy mutates, it will be likely that ambiguous players will mutate to precise players rapidly. Choosing the variables Population ratios and χ 2 -tests of homogeneity Before experimenting we have to decide which variables and which constants to use. Since there is a wide variety of choices to be made, it would be interesting to see which actually make a big difference. Possibly, one of these is the initial ratio ambiguous:precise strategies that we can use before iterating. The plot (figure 4) shows that there is a chance that under these conditions, but while changing β, it does not matter what the initial population is. We consider three extremes: 1. Population A where the ratio is 50:50, population B where the ratio is 90:10 and population C where the ratio is 10:90. For this we run 3 3 Pearson χ 2 tests of homogeneity to check if the population ratios are indeed similar. We have one degree of freedom and the null hypotheses are: H 0 : P ratio ambiguous:precise players of population A = P ratio ambiguous:precise players of population B H 0 : P ratio ambiguous:precise players of population A = P ratio ambiguous:precise players of population C H 0 : P ratio ambiguous:precise players of population B = P ratio ambiguous:precise players of population C H A : at least one of the H 0 -statements is false We can now do a Pearson s χ 2 -test of homogeneity with these results. The results form the observed frequency count in the formula: O r,c where r stands for 1 The notions of payoff and utility are interchangeable 10
Figure 4: Ratio ambiguous:precise players of population A, B and C with β = 1.0, 0.9, 0.2 11
Table 1: Results of experiment plotratios Population A Population B Population C preccost=1.0 N = 1000 N = 1000 N = 1000 # Ambiguous players 26 20 22 # Precise players 974 980 978 preccost=0.9 # Ambiguous players 270 302 293 # Precise players 730 698 707 preccost=0.2 # Ambiguous players 572 572 593 # Precise players 428 428 407 Note that results are also based on probability of mutation so results will differ each experiment. the population and c for either ambiguous or precise. We care for a significance level of p = 0.95. We will do one for all three hypotheses for all three β s in the same way. For the results see appendix E. We can conclude that all populations are homogeneous in ambiguous:precise player strategy ratio after 100 iterations. We assume that 100 iterations are enough to produce a stabilised ambiguous:precise ratio in the setup, as the plots demonstrate that the ratio does not majorly change after 100 iterations. 12
Furthermore, two variables are interesting in particular: the β payoff and the probability of mutation. β-payoff for precise senders The β-payoff is the payoff that a precise sender receives after successful signalling. It is chosen to only make precise sending costly (and not also precise receiving) because of the assumption that in natural language it is not as costly to hear precise language and interpret it, as to produce the precise language itself. The following piece of code shows where the β-value 2 gets awarded as a payoff. def game( events, sender, receiver, precpayoff): # pr ecpa yoff between 0 and 1 """The signalling game""" for x in range (0, 1): event = random. choice( events) signal = sender. picksignal( event) action = receiver. pickaction( signal) # i f s u c c e s s f u l signalling, d i s t r i b u t e payoffs if event == action: # payoff depends on s t r a t e g y t y p e if sender. strategytype == prectype: sender. payoffs += precpayoff else: sender. payoffs += 1 if receiver. strategytype == prectype: receiver. payoffs += 1 else: receiver. payoffs += 1 sender. nrofgamesplayed += 1 receiver. nrofgamesplayed += 1 Probability of mutation There are multiple ways one could control the likelihood of a changing player s strategy. For example, one could set a bound: if a player s payoff, divided by the games played 3, is lower than this bound, the player would change its strategy. To illustrate: one could set this bound to 0.9. After each iteration, a player s mean payoff can change. If its mean payoff starts off as perfect (1.0), but eventually 2 in the code: precpayoff 3 in the code: ratios for the sender and ratior for the receiver 13
crosses the bound after a few iterations, its strategy will mutate and the mean payoff will be reset. This surely is a possible way to control the mutation. However, intuitively a language user does not change its strategy so explicitly. It is why there is chosen to take a slightly different approach. In the experiments of this research there needs to be inserted a probability of mutation -value 4. After each signalling game, a floating number is randomly chosen between zero and this probability-value. If this number is higher than the mean players payoff, the players strategy mutates. Because this probability-value influences the eventual ratio ambiguous:precise signallers, we will experiment with different values to see whether the amount of mutation will decrease or not after a certain point in iteration. The following piece of code is to establish an amount of games 5 between the population s players. Here it is shown how the mean players payoffs and the probability-value influence the probability of mutation. for i in range (0, nrofgames): amountofmutation = 0 for sender, receiver in random pairs of ( players): game( events, sender, receiver, precpayoff) ratios = sender. payoffs/ float( sender. nrofgamesplayed) ratior = receiver. payoffs/ float( receiver. nrofgamesplayed) if random. uniform(0, probability) > ratios: amountofmutation += 1 # a mutation t a k e s place if sender. strategytype == ambtype: sender. strategytype = prectype else: sender. strategytype = ambtype sender.payoffs = 0 # a f t e r mutation, a l l payoff i s l o s t sender. nrofgamesplayed = 0 if random. uniform(0, probability) > ratior: if receiver. strategytype == ambtype: receiver. strategytype = prectype else: receiver. strategytype = ambtype receiver.payoffs = 0 # a f t e r mutation, a l l payoff i s l o s t receiver. nrofgamesplayed = 0 4 in the code: probability 5 in the code: nrofgames 14
Results Varying the β-value In figure 5 a plot is shown that represents the experiment with a changing β-value. The probability of mutation is 1. The final percentage of ambiguous players that is left after 100 iterations, in a population of 1000 players with an initial 50:50 ratio of ambiguous:precise players is shown. This is the mean out of three experiments with the function makeutilityplot() 6, to normalise the results, for they may differ after each experiment because the results depend on probabilities. The value P, that determines the probability of mutation was 1.0. The β-value is between 0 and 1, each time increasing with 0.01. When the β-value is 0, it means that precise senders do not get any payoff. In this case, the percentage of ambiguous players increases from 50% to 64%. Precise strategies still provide for a significant amount of profit (successful precise receiving still accounts for a payoff of 1) to make up for 36% of the population when they do not get any payoff for precise sending. As the β-value comes closer to 1, the final percentage ambiguous plays drops more rapidly. The most severe change in the ambiguous:precise players happens between β = 0.9 and β = 1.0. The results of this part of the plot are also shown in figure 6. As shown, when the β-value is 1, which means no payoff for the precise signaller, there are still ambiguous players left: approximately 4%. 6 see appendix C 15
Figure 5: The percentage ambiguous signallers after 100 iterations with varying β-values from 0 to 1 Figure 6: The percentage ambiguous signallers after 100 iterations with varying β-values from 0.9 to 1 16
Figure 7: Upper plot: Percentage of a 50:50 population that mutates with three different probabilities of mutation, P, during 50 iterations. Lower plot: Percentage ambiguous signallers during 50 iterations Varying the probability of mutation In figure 7, in the upper plot, the percentage of a population which strategy mutates after signalling, for iteration 0 to 50, is shown. The percentages are the mean out of three experiments. This is done to take into account that the mutation is based on a probability, which makes the results of each experiment different. Taking the mean out of 3 experiments makes the result more clear. Both the blue, green and red line represent a 50:50 ratio and every line represents a different mutation probability, P. Three probabilities are considered: 1.1, 1.0 and 0.9. As expected, the amount of players that still mutates after a few games is higher when the probability of mutation is higher. 17
However, it seems that when P = 1.1, the amount of mutation stabilises after 10 iterations, whereas the amount of mutation keeps dropping when P < 1.0. It is trivial that the ratio ambiguous:precise signallers stabilises when there s almost no mutation left. But when the probability of mutation is higher than 1, it actually means that, although there could be perfect signalling, there s always a chance of mutating. This in contrast to when the probability is 1.0 or lower, where a perfect signaller would never mutate its strategy because the outcome of a random float between 0 and 1.0 will never be higher than 1.0 7. This is why it is interesting to investigate whether the ratio ambiguous:precise signallers still changes when the amount of mutation does not drop. The results of the ratio ambiguous:precise players that belong to the mutation experiment in the upper plot are demonstrated in the lower plot of figure 7. 7 this number is determined by random.uniform(0,probability) 18
Conclusion In this research experiments have been done with evolutionary signalling games. Populations of 1000 players with an initial 50:50 ratio between ambiguous and precise signalling strategies have played 100 signalling games. After each signalling game, a player s strategy had the possibility that its strategy would mutate, depending on the probability of mutation(value P ) and the β-payoff that was assigned to successful signalling by a precise sender. Before experimenting with the cost, an experiment has been done to investigate whether the initial ratio ambiguous:precise signallers of a population, before playing the games, was of significant importance for the resulting ratio ambiguous:precise players. Three populations with three different initial ambiguous:precise players ratios (namely 10:90, 50:50 and 90:10) have been examined with again three experiments with different β-values. A plot of these experiments suggested that the final population ratio ambiguous:precise players is not dependent on the initial ratio. Pearson s χ 2 -tests of homogeneity showed that the difference between two populations is indeed not significant. After this result, it was possible to experiment only with 50:50 initial population ratios. These experiments showed that the ambiguous strategy that was chosen is welladapted for surviving in evolutionary signalling games. The final percentage ambiguous players in a population is higher when the β-value for precise sending is lower. However, even when there is no special payoff for precise senders(when β = 1), ambiguous signallers survive when the probability of mutation P = 1. This is a small percentage however, approximately 4%. An interpretation is that ambiguous signallers still exist because they still receive a significant amount of payoff when playing a signalling game with a precise signaller. As calculated in the method section, the mean payoff for a game with one precise and one ambiguous player is relatively high, especially when the precise player is the sender and the ambiguous player is the receiver. This is mainly because of the choice to use an ambiguous strategy where only half of the signals are assigned to two meanings and half of the signals is only assigned to one meaning. It turns out that the resulting percentage of ambiguous players after 100 iterations is probably stable, because the percentage of the population that mutates drops as the percentage of ambiguous players drops too. In the case where the resulting amount of ambiguous players is 4%, the amount that mutates is also 4%. 19
Discussion It is found costly precise sending in evolutionary signalling games help ambiguous strategies to survive. However, it is not yet clear to what extend this conclusion can generalise. The games are based on a few assumptions. For instance, it is not guaranteed that in natural language, precise language is always costlier than ambiguous language. This however can be investigated more thoroughly in future research. Signalling games in general are a drastically simplified representation of communication in natural language. A lot of properties of natural language use have not been taken into account. In the light of this research it would have been a better model if multiple experiments were done with different ambiguous and precise player strategies. An obvious strategy that would be conducive for this matter is a strategy that is partly ambiguous, partly precise. This is because natural language users also do not strictly use one or the other. Aside from that, there are a lot more player strategies one could think of that are used in natural language but not in this research. An example has already been given in the introduction of this thesis, namely a strategy with synonymy. Another choice that influenced the results was that players in a population randomly are assigned to play a signalling game with one another. It would be interesting to see what would happen if instead players would be put in a grid. In this grid they could e.g., have a higher probability of playing against their neighbours which would yield a more realistic way of encountering between interlocutors. For future research it is recommended to examine the discussed assumptions. Aside from that it is interesting to investigate what the impact of the availability of context would have in the signalling games. For players to invoke context while interpreting a signal, would intuitively be in favour of ambiguous strategies: context can help players to choose the right meaning to an ambiguous signal. 20
References [1] David Catteeuw and Bernard Manderick. The limits and robustness of reinforcement learning in lewis signalling games. Connection Science, 26(2):161 177, 2014. [2] Judith Degen, Michael Franke, and Gerhard Jäger. Cost-based pragmatic inference about referential expressions. In Proceedings of the 35th Annual Conference of the Cognitive Science Society, pages 376 381, 2013. [3] Gerhard Jäger. Evolutionary game theory and typology: A case study. Language, pages 74 109, 2007. [4] Gerhard Jäger. Evolutionary stability conditions for signaling games with costly signals. Journal of theoretical biology, 253(1):131 141, 2008. [5] Simon Kirby. Fitness and the selective adaptation of language. Approaches to the evolution of language: Social and cognitive bases, pages 359 383, 1998. [6] David Lewis. Convention: A philosophical study. John Wiley & Sons, 2008. [7] Roland Mühlenbernd and Michael Franke. Meaning, evolution, and the structure of society. Unpublished manuscript, 2014. [8] Carlos Santana. Ambiguity in cooperative signaling. 81(3):398 422, July 2014. [9] Brian Skyrms. Signals: Evolution, learning, and information. Oxford University Press, 2010. [10] Joel Sobel. Signaling games. In Computational Complexity, pages 2830 2844. Springer, 2012. [11] Peter E. Trapa and Martin A. Nowak. Nash equilibria for an evolutionary language game. Journal of mathematical biology, 41(2):172 188, 2000. [12] George Kingsley Zipf. Human behavior and the principle of least effort. 1949. 21
Appendices 22
Appendix A: Classes Ambiguous Player & Precise Player import random class AmbiguousStrategy: def picksignal( self, event): if event == a : signal = s1 elif event == b : signal = s1 elif event == c : signal = s3 elif event == d : signal = s3 return signal def pickaction( self, signal): if signal == s1 : action = random.choice([ a, b ]) elif signal == s2 : action = b elif signal == s3 : action = action = random.choice([ c, d ]) elif signal == s4 : action = d return action class PreciseStrategy: def picksignal( self, event): if event == a : signal = s1 elif event == b : signal = s2 elif event == c : signal = s3 elif event == d : signal = s4 return signal def pickaction( self, signal): if signal == s1 : action = a elif signal == s2 : 23
action = b elif signal == s3 : action = c elif signal == s4 : action = d return action 24
Appendix B: Class Evolutionary Signalling Games This is the part where the signalling takes place. The function needs the set of events, two players(sender and receiver) and the precpayoff 8. The signals are known to the players because they are defined in their classtype (ambiguousplayer of preciseplayer). def game( events, sender, receiver, precpayoff): """The signalling game""" for x in range (0, 1): event = random. choice( events) signal = sender. picksignal( event) action = receiver. pickaction( signal) if event == action: # payoff depends on s t r a t e g y t y p e if sender. strategytype == prectype: sender. payoffs += precpayoff else: sender. payoffs += 1 if receiver. strategytype == prectype: receiver. payoffs += 1 else: receiver. payoffs += 1 sender. nrofgamesplayed += 1 receiver. nrofgamesplayed += 1 8 precpayoff=β 25
Appendix C: Code for experiments from f u t u r e import division from PreciseStrategy import PreciseStrategy from AmbiguousStrategy import AmbiguousStrategy from Player import Player import random from random import shuffle from itertools import izip import matplotlib. pyplot as plt from numpy import arange This file contains code for experimenting with ambiguous strategies in evolutionary signalling games events = [ a, b, c, d ] # = actions signals = [ s1, s2, s3, s4 ] # the available s t r a t e g i e s prectype = PreciseStrategy() ambtype = AmbiguousStrategy() # 3 l i s t s of players with d i f f e r e n t strategy r a t i o s players5050 = list() players1090 = list() players9010 = list() players50502 = list() players10902 = list() players90102 = list() players50503 = list() players10903 = list() players90103 = list() for i in range (500): players5050. append( Player( prectype)) 26
players5050.append(player(ambtype)) players50502. append( Player( prectype)) players50502. append( Player( ambtype)) players50503. append( Player( prectype)) players50503. append( Player( ambtype)) for i in range (100): players1090. append( Player( prectype)) players9010.append(player(ambtype)) players10902. append( Player( prectype)) players90102. append( Player( ambtype)) players10903. append( Player( prectype)) players90103. append( Player( ambtype)) for i in range (900): players1090.append(player(ambtype)) players9010. append( Player( prectype)) players10902. append( Player( ambtype)) players90102. append( Player( prectype)) players10903. append( Player( ambtype)) players90103. append( Player( prectype)) random. shuffle( players5050) random. shuffle( players9010) random. shuffle( players1090) random. shuffle( players50502) random. shuffle( players90102) random. shuffle( players10902) random. shuffle( players50503) random. shuffle( players90103) random. shuffle( players10903) # For when only 50:50 populations are needed def create5050population (): 27
players5050 = list() for i in range (500): players5050. append( Player( prectype)) players5050.append(player(ambtype)) random. shuffle( players5050) return players5050 # Decide which players w i l l play against each other def random pairs of ( players): """ Return all of players as random pairs.""" # copy player l i s t players = list( players ) # s h u f f l e the new player l i s t in place random.shuffle(players) # y i e l d the s h u f f l e d players, 2 at a time player iter = iter( players) return izip( player iter, player iter) # Let a population play against each other for nrofgames times def playmanygames( players, events, nrofgames, probability, precpayoff): """Play multiple games""" percentages = range( nrofgames +1) nrofambplayers = 0 nrofprecplayers = 0 AmbPayoffs = 0 PrecPayoffs = 0 for x in players: if x. strategytype == ambtype: nrofambplayers += 1 AmbPayoffs = AmbPayoffs + x. payoffs else: 28
nrofprecplayers += 1 PrecPayoffs = PrecPayoffs + x. payoffs percentages [0] = \ ( nrofambplayers/ float( nrofambplayers + nrofprecplayers )) 100 mutationlist = list() for i in range (0, nrofgames): amountofmutation = 0 for sender, receiver in random pairs of ( players): game( events, sender, receiver, precpayoff) ratios = sender. payoffs/ float( sender. nrofgamesplayed) ratior = receiver. payoffs/ float( receiver. nrofgamesplayed) if random. uniform(0, probability) > ratios: amountofmutation += 1 # a mutation t a k e s place if sender. strategytype == ambtype: sender. strategytype = prectype else: sender. strategytype = ambtype sender.payoffs = 0 # a f t e r mutation, lose payoff sender. nrofgamesplayed = 0 if random. uniform(0, probability) > ratior: if receiver. strategytype == ambtype: receiver. strategytype = prectype else: receiver. strategytype = ambtype receiver. payoffs = 0 receiver. nrofgamesplayed = 0 nrofambplayers = 0 nrofprecplayers = 0 AmbPayoffs = 0 PrecPayoffs = 0 for x in players: if x. strategytype == ambtype: nrofambplayers += 1 AmbPayoffs = AmbPayoffs + x. payoffs else: nrofprecplayers += 1 PrecPayoffs = PrecPayoffs + x. payoffs percentages[i+1] = \ ( nrofambplayers/ float( nrofambplayers + nrofprecplayers )) 100 mutationlist. append( float( amountofmutation /10)) 29
return percentages, mutationlist # Show t h a t the same % amb players with 3 d i f f e r e n t i n i t i a l population # r a t i o s and 3 d i f f e r e n t precpayoffs w i l l arise def makeratioplot( players5050, players9010, players1090, events): """ Plots 3 kinds of populations with 3 different utility functions""" percentages5050, mutationlist = \ playmanygames( players5050, events, 100, 1, 1) percentages9010, mutationlist = \ playmanygames( players9010, events, 100, 1, 1) percentages1090, mutationlist = \ playmanygames( players1090, events, 100, 1, 1) percentages50502, mutationlist = \ playmanygames( players50502, events, 100, 1, 0.9) percentages90102, mutationlist = \ playmanygames( players90102, events, 100, 1, 0.9) percentages10902, mutationlist = \ playmanygames( players10902, events, 100, 1, 0.9) percentages50503, mutationlist = \ playmanygames( players50503, events, 100, 1, 0.2) percentages90103, mutationlist = \ playmanygames( players90103, events, 100, 1, 0.2) percentages10903, mutationlist = \ playmanygames( players10903, events, 100, 1, 0.2) print percentages5050[ 1], percentages9010[ 1], percentages1090[ 1],\ percentages50502[ 1], percentages90102[ 1],percentages10902[ 1],\ percentages50503[ 1], percentages90103[ 1], percentages10903[ 1] t = arange(0., 101) fig = plt. figure(1) plt. subplot (311) plt. title("3 amb: prec strategy ratios with \ 3 different precpayoffs ( Beta value)") l1,l2, l3 = plt. plot(t, percentages5050, \ 30
t, percentages9010, t, percentages1090) plt. ylabel( % amb players, Beta =1.0 ) # p l t. xlabel ( # i t e r a t i o n s ) plt. subplot (312) plt. plot(t, percentages50502, t, percentages90102, t, percentages10902) plt. ylabel( % amb players, Beta =0.9 ) # p l t. xlabel ( # i t e r a t i o n s ) plt. subplot (313) plt. plot(t, percentages50503, t, percentages90103, t, percentages10903) plt. ylabel( % amb players, Beta =0.2 ) plt. xlabel( # iterations ) lines = l1,l2,l3 labels = [ 50:50, 90:10, 10:90 ] plt. figlegend( lines, labels, loc = upper center,\ ncol =5, labelspacing =0. ) fig. patch. set facecolor( white ) plt.show() # Make a p l o t of the ending amb: prec r a t i o with d i f f e r e n t preccosts def makeutilityplot( events, nrofgames, probability, precpayoffs): costpercentages = list() for i in precpayoffs: players = create5050population() precpayoff = float(i/10) costpercentage, mutationlist = playmanygames( players,\ events, nrofgames, probability, precpayoff) costpercentages. append( costpercentage[ 1]) return costpercentages def plotdifferentpreccosts( events): precpayoffs = list() for i in range(0,1000,10): precpayoffs. append(i / 100.00) 31
collectedcostpercentages = list() for i in range(0,3): costpercentages = makeutilityplot( events, 100, 1, precpayoffs) collectedcostpercentages. append( costpercentages) meanpercentages = [ sum(e)/ len(e) for e in zip( collectedcostpercentages)] fig = plt. figure(1) plt. title(" Amount of ambiguous signallers after 100 iterations") precpayoffs = [x / 10 for x in precpayoffs] l1 = plt. plot( precpayoffs, meanpercentages) plt. ylabel( % amb players (n =1000) after taking the mean of 3 experiments plt. xlabel( payoff for sender with precise strategy ) fig. patch. set facecolor( white ) plt.show() plotdifferentpreccosts( events) # Plot the % of the population t h a t mutates during i t e r a t i n g def plotmutations( events, players1, players2, players3): collectedmutations = list() collectedmutations2 = list() collectedmutations3 = list() for i in [ players1, players2, players3]: costpercentage, mutationlist = \ playmanygames(i, events, 100, 1.1, 0.8) collectedmutations. append( mutationlist) meanmutations = [ sum(e)/ len(e) for e in zip( collectedmutations)] for i in [ players5050, players50502, players50503]: costpercentage, mutationlist = \ playmanygames(i, events, 100, 1.1, 0.8) collectedmutations2. append( mutationlist) meanmutations = [ sum(e)/ len(e) for e in zip( collectedmutations)] for i in [ players1090, players10902, players10903]: costpercentage, mutationlist = \ playmanygames(i, events, 100, 1.1, 0.8) collectedmutations3. append( mutationlist) meanmutations = [ sum(e)/ len(e) for e in zip( collectedmutations)] meanmutations2 = [ sum(e)/ len(e) for e in zip( collectedmutations2)] meanmutations3 = [ sum(e)/ len(e) for e in zip( collectedmutations3)] t = arange(0., 100) 32
fig = plt. figure(1) plt. title("% of 3 different populations that mutates \ in a signalling round with probability variable =1.1 and precpayoff =0 l1,l2, l3 = plt. plot(t, meanmutations,\ t, meanmutations2, t, meanmutations3 ) plt. ylabel( % mutations ( the mean of 3 experiments) ) plt. xlabel( # iterations ) fig. patch. set facecolor( white ) lines = l1,l2,l3 labels = [ 10:90, 50:50, 90:10 ] plt. figlegend( lines, labels, loc = upper center,\ ncol =5, labelspacing =0. ) plt.show() 33
Appendix D: Plots 34
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Appendix E: χ 2 -test for homogeneity of populations We have done nine Pearson s χ 2 tests of homogeneity with RStudio to test the equality of population ratio ambiguous:precise strategies between three different initial populations with three different β-values. These were the observed values: Hence, these nine tests are done with following these formulas: Degrees of Freedom: DF = (r 1) (c 1) = 1 1 = 1 Expected frequency counts: E r,c = (n r n c )/n Test statistic: 2 = [(O r,c E r,c ) 2 /E r,c ] β = 1.0: Population A & B: χ 2 = 0.80103, df = 1, p-value = 0.3708 Population B & C: χ 2 = 0.097281, df = 1, p-value = 0.7551 Population A & C: χ 2 = 0.34153, df = 1, p-value = 0.5589 β = 0.9: Population A & B: χ 2 = 2.5073, df = 1, p-value = 0.1133 Population B & C: χ 2 = 0.19379, df = 1, p-value = 0.6598 Population A & C: χ 2 = 1.3077, df = 1, p-value = 0.2528 β = 0.2: 38
Population A & B: χ 2 = 0, df = 1, p-value = 1 Population B & C: χ 2 = 0.90668, df = 1, p-value = 0.341 Population A & C: χ 2 = 0.90668, df = 1, p-value = 0.341 CV = 3.84, so all X1 2 s are smaller than the critical value of 3.84. Hence, we cannot reject any of the null hypotheses. We can conclude that all populations are homogeneous. Even when we do a Yates correction, still all the null hypotheses cannot be rejected. We proceed with a population ratio of 50:50. 39