# The Evolution of Ambiguity in Sender—Receiver Signaling Games

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Structure of the Article

## 2. The Game Models

#### 2.1. Lewis Signaling Game

#### 2.2. Context-Signaling Game

- $Pr\left({t}_{1}\phantom{\rule{3.33333pt}{0ex}}\right|{c}_{1})=2\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}3$, $Pr\left({t}_{1}\phantom{\rule{3.33333pt}{0ex}}\right|{c}_{2})=0$
- $Pr\left({t}_{2}\phantom{\rule{3.33333pt}{0ex}}\right|{c}_{1})=1\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}3$, $Pr\left({t}_{2}\phantom{\rule{3.33333pt}{0ex}}\right|{c}_{2})=1\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}3$
- $Pr\left({t}_{3}\phantom{\rule{3.33333pt}{0ex}}\right|{c}_{1})=0$, $Pr\left({t}_{3}\phantom{\rule{3.33333pt}{0ex}}\right|{c}_{2})=2\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}3$

#### 2.3. Context Bottleneck Game

## 3. Formal and Computational Analysis

- $EU(\gamma ,\gamma )\ge EU({\gamma}^{\prime},\gamma )$ for all ${\gamma}^{\prime}\ne \gamma $
- If $EU(\gamma ,\gamma )=EU({\gamma}^{\prime},\gamma )$ for some ${\gamma}^{\prime}\ne \gamma $, then $EU(\gamma ,{\gamma}^{\prime})>EU({\gamma}^{\prime},{\gamma}^{\prime})$

#### 3.1. Strategy Spaces and Equilibria

#### 3.2. Emergence Rates under Evolutionary Dynamics

## 4. Online Experiments

**Hypothesis**

**1.**

**Hypothesis**

**2.**

**Hypothesis**

**3.**

**Hypothesis**

**4.**

#### 4.1. Experimental Setup

- player 1 is sender, information state is ${t}_{2}$;
- player 2 is sender, information state is ${t}_{3}$;
- player 1 is sender, information state is ${t}_{1}$;
- player 2 is sender, information state is ${t}_{2}$;
- player 1 is sender, information state is ${t}_{3}$;
- player 2 is sender, information state is ${t}_{1}$.

#### 4.2. Experimental Results

#### 4.3. Discussion

## 5. Conclusions

## 6. Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CS game | context-signaling game |

LS game | Lewis signaling game |

CB game | context bottleneck game |

$EU$ | expected utility |

${U}_{C}$ | communicative utility |

EGT | evolutionary game theory |

ESS | evolutionarily stable strategy |

PDI | pairwise difference imitation |

CoS | communicative success |

PS | perfect signaling |

PA | perfect ambiguity |

nPA | non-perfect ambiguity |

## Appendix A. Pairwise Differential Imitation (PDI) Dynamics

## Appendix B. Experimental Procedure

- In this experiment you will play a communication game with an other participant for a number of 30 rounds.
- In each round you both can score 10 points if you play successfully, otherwise you both receive 0 points.
- Your final total score will be converted into real money (100 points = 1£) and added to your participation fee.
- Please take your time and play carefully. Press ’Next’ to go to the video tutorial (<2 min) that explains how to play the game.

**Figure A2.**Screenshots of an exemplary interaction round for the LS game, with the green agent as sender and the blue agent as receiver. (

**a**) Initial perspective of the green agent in sender role. Her private information state is ’banana’ (alternatives: ’apple’, ’grapes’), and she has to pick a signal, $, & or §. (

**b**) Perspective of the blue agent (receiver role) after the sender has picked signal &. He cannot see the information state of the green agent and has to guess an information state as response: ’apple’, ’banana’ or ’grapes’. (

**c**) Perspective of both agents after the receiver has picked ’grapes’ as response. Communication failed in this example, and both don’t score.

**Figure A3.**Screenshots of the final screen of an exemplary interaction round for the CS game. The contextual cue is presented as a disjunction of two information states, of which one is true.

## References

- Lewis, D. Convention. A Philosophical Study; Blackwell: Cambridge, MA, USA, 1969. [Google Scholar]
- Barrett, J.A. Numerical Simulations of the Lewis Signaling Game: Learning Strategies, Pooling Equilibria, and the Evolution of Grammar; Technical Report; Institute for Mathematical Behavioral Sciences, University of California: Irvine, UK, 2006. [Google Scholar]
- Skyrms, B. Signals: Evolution, Learning and Information; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Huttegger, S.M.; Zollman, K.J.S. Signaling Games: Dynamics of Evolution and Learning. In Language, Games, and Evolution; Benz, A., Ebert, C., Jäger, G., van Rooij, R., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 160–176. [Google Scholar]
- Wärneryd, K. Cheap Talk, Coordination, and Evolutionary Stability. Games Econ. Behav.
**1993**, 5, 532–546. [Google Scholar] [CrossRef] - Huttegger, S.M. Evolution and the Explanation of Meaning. Philos. Sci.
**2007**, 74, 1–27. [Google Scholar] [CrossRef] - Santana, C. Ambiguity in Cooperative Signaling. Philos. Sci.
**2014**, 81, 398–422. [Google Scholar] [CrossRef] - Mühlenbernd, R. Evolutionary stability of ambiguity in context-signaling games. Synthese
**2021**, 198, 11725–11753. [Google Scholar] [CrossRef] - Skyrms, B. Evolution of the Social Contract; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Skyrms, B.; Pemantle, R. A dynamic model of social network formation. Proc. Natl. Acad. Sci. USA
**2000**, 97, 9340–9349. [Google Scholar] [CrossRef][Green Version] - Zollman, K.J.S. Talking to Neighbors: The Evolution of Regional Meaning. Philos. Sci.
**2005**, 72, 69–85. [Google Scholar] [CrossRef][Green Version] - Hofbauer, J.; Huttegger, S.M. Feasibility of communication in binary signaling games. J. Theor. Biol.
**2008**, 245, 843–849. [Google Scholar] [CrossRef] - Pawlowitsch, C. Why Evolution does not always lead to an optimal signaling system. Games Econ. Behav.
**2008**, 63, 203–226. [Google Scholar] [CrossRef] - Barrett, J.A.; Zollman, K.J.S. The Role of Forgetting in the Evolution and Learning of Language. J. Exp. Theor. Artif. Intell.
**2009**, 21, 293–309. [Google Scholar] [CrossRef] - Mühlenbernd, R. Learning with Neighbours. Synthese
**2011**, 183, 87–109. [Google Scholar] [CrossRef] - Mühlenbernd, R.; Franke, M. Meaning, evolution and the structure of society. In Proceedings of the European Conference on Social Intelligence, Barcelona, Spain, 3–5 November 2014; Herzig, A., Lorini, E., Eds.; Volume 1283, pp. 28–39. [Google Scholar]
- Mühlenbernd, R.; Nick, J. Language change and the force of innovation. In Pristine Perspectives on Logic, Language, and Computation; Katrenko, S., Rendsvig, K., Eds.; Springer: Heidelberg, Germany; New York, NY, USA, 2014; Volume 8607, pp. 194–213. [Google Scholar]
- Mühlenbernd, R.; Enke, D. The grammaticalization cycle of the progressive—A game-theoretic analysis. Morphology
**2017**, 27, 497–526. [Google Scholar] [CrossRef] - Mühlenbernd, R. The change of signaling conventions in social networks. AI Soc.
**2019**, 34, 721–734. [Google Scholar] [CrossRef] - Macy, M.W.; Flache, A. Learning dynamics in social dilemmas. Proc. Natl. Acad. Sci. USA
**2002**, 99, 7229–7236. [Google Scholar] [CrossRef] [PubMed][Green Version] - Skyrms, B. The Stag Hunt and the Evolution of Social Structure; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Nowak, M.A. Five rules for the evolution of cooperation. Science
**2006**, 314, 1560–1563. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lorini, E.; Mühlenbernd, R. The long-term benefits of following fairness norms under dynamics of learning and evolution. Fundam. Inform.
**2018**, 158, 121–148. [Google Scholar] [CrossRef] - LiCalzi, M.; Mühlenbernd, R. Categorization and cooperation across games. Games
**2019**, 10, 5. [Google Scholar] [CrossRef][Green Version] - Harré, M. Utility, Revealed Preferences Theory, and Strategic Ambiguity in Iterated Games. Entropy
**2017**, 19, 201. [Google Scholar] [CrossRef][Green Version] - Blume, A.; DeJong, D.V.; Kim, Y.G.; Sprinkle, G.B. Evolution of Communication with Partial Common Interest. Games Econ. Behav.
**2001**, 37, 79–120. [Google Scholar] [CrossRef][Green Version] - Bruner, J.; O’Connor, C.; Rubin, H.; Huttegger, S.M. David Lewis in the lab: Experimental results on the emergence of meaning. Synthese
**2018**, 195, 603–621. [Google Scholar] [CrossRef] - Rubin, H.; Bruner, J.; O’Connor, C.; Huttegger, S.M. Communication without common interest: A signaling experiment. Stud. Hist. Philos. Sci. Part C Stud. Hist. Philos. Biol. Biomed. Sci.
**2020**, 83, 101295. [Google Scholar] [CrossRef] - Blume, A.; Lai, E.; Lim, W. Strategic information transmission: A survey of experiments and theoretical foundations. In Handbook of Experimental Game Theory; Capra, C.M., Croson, R., Rigdon, M., Rosenblat, T., Eds.; Edward Elgar Publishing: Cheltenham, UK; Northampton, MA, USA, 2020; pp. 311–347. [Google Scholar]
- Rohde, H.; Seyfarth, S.; Clark, B.; Jaeger, G.; Kaufmann, S. Communicating with cost-based implicature: A game-theoretic approach to ambiguity. In Proceedings of the 16th Workshop on the Semantics and Pragmatics of Dialogue, Paris, France, 19–21 September 2012. [Google Scholar]
- Schumann, A. Payoff Cellular Automata and Reflexive Games. J. Cell. Autom.
**2014**, 9, 287–313. [Google Scholar] - Schumann, A. Towards Context-Based Concurrent Formal Theories. Parallel Process. Lett.
**2015**, 25, 1540008. [Google Scholar] [CrossRef] - Mertens, J.F.; Neyman, A. Stochastic games. Internatioanl J. Game Theory
**1981**, 10, 53–66. [Google Scholar] [CrossRef] - Hilbe, C.; Štěpán, Š.; Chatterjee, K.; Nowak, M.A. Evolution of cooperation in stochastic games. Nature
**2018**, 559, 246–249. [Google Scholar] [CrossRef] [PubMed] - Jäger, G. Evolutionary Game Theory and Typology. A Case Study. Language
**2007**, 83, 74–109. [Google Scholar] [CrossRef] - Deo, A. The semantic and pragmatic underpinnings of grammaticalization paths: The progressive to imperfective shift. Semant. Pragmat.
**2015**, 8, 1–52. [Google Scholar] [CrossRef][Green Version] - Bruner, J.; O’Connor, C.; Rubin, H. Experimental economics for philosophers. In Methodological Advances in Experimental Philosophy; Fischer, M.C.E., Ed.; Bloomsbury Academic: New York, NY, USA, 2019. [Google Scholar]
- Spence, M. Job market signaling. Q. J. Econ.
**1973**, 87, 355–374. [Google Scholar] [CrossRef] - Farrell, J.; Rabin, M. Cheap Talk. J. Econ. Perspect.
**1996**, 10, 103–118. [Google Scholar] [CrossRef] - Jäger, G. Applications of Game Theory in Linguistics. Lang. Linguist. Compass
**2008**, 2/3, 408–421. [Google Scholar] - Mühlenbernd, R.; Quinley, J. Language change and network games. Lang. Linguist. Compass
**2017**, 11, e12235. [Google Scholar] [CrossRef] - Grafen, A. Biological signals as handicaps. J. Theor. Biol.
**1990**, 144, 517–546. [Google Scholar] [CrossRef] - Maynard Smith, J. The concept of information in biology. Philos. Sci.
**2000**, 67, 177–194. [Google Scholar] [CrossRef] - Maynard Smith, J.; Price, G. The Logic of Animal Conflict. Nature
**1973**, 246, 15–18. [Google Scholar] [CrossRef] - Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982. [Google Scholar]
- Nowak, M.A.; Krakauer, D.C. The evolution of language. Proc. Natl. Acad. Sci. USA
**1999**, 96, 8028–8033. [Google Scholar] [CrossRef] [PubMed][Green Version] - Taylor, P.D.; Jonker, L.B. Evolutionarily Stable Strategies and Game Dynamics. Math. Biosci.
**1978**, 40, 145–156. [Google Scholar] [CrossRef] - Balkenborg, D.; Schlag, K.H. Evolutionarily stable sets. Int. J. Game Theory
**2001**, 29, 571–595. [Google Scholar] [CrossRef] - Izquierdoy, L.R.; Izquierdoz, S.S.; Sandholm, W.H. An Introduction to ABED: Agent-Based Simulation of Evolutionary Game Dynamics. Games Econ. Behav.
**2019**, 118, 434–462. [Google Scholar] [CrossRef] - Skyrms, B. Signals, evolution and the explanatory power of transient information. Philos. Sci.
**2002**, 69, 407–428. [Google Scholar] [CrossRef][Green Version] - Aumann, R. Nash equilibria are not self-enforcing. In Economic Decision Making, Games, Econometrics and Optimization; Gabzewicz, J.J., Richard, J.F., Wolsey, L.A., Eds.; North Holland: Amsterdam, The Netherlands, 1990; pp. 201–206. [Google Scholar]
- Roth, A.E.; Erev, I. Learning in Extensive-Form Games: Experimental Data and Simple Dynamic Models in the Intermediate Term. Games Econ. Behav.
**1995**, 8, 164–212. [Google Scholar] [CrossRef] - Fudenberg, D.; Levine, D.K. The Theory of Learning in Games; MIT Press: Cambridge, MA, USA, 1998. [Google Scholar]

**Figure 2.**A perfect signaling system of the CS game is shown in (

**a**), and a perfect ambiguous system of the CS game is shown in (

**b**). Both achieve an expected utility of 1.

**Figure 3.**The two perfect ambiguous systems of the CB game are shown in (

**a**,

**b**), both of which achieve an expected utility of 1. Two (of 12) exemplary non-perfect but evolutionarily stable pooling systems of the CB game are shown in (

**c**,

**d**), both of which achieve an expected utility of $\frac{5}{6}$.

**Figure 4.**Communicative success (CoS) rates of the experiments. (

**a**) shows the CoS rates over initial 6, final 6 and all rounds, averaged over all participants for each game type. (

**b**–

**f**) show the CoS rates over blocks of all participant pairs for Sessions I to V, respectively.

**Figure 5.**Frequency of types of communication systems that emerged in the laboratory experiments (

**a**) and in the simulation runs under evolutionary dynamics (

**b**). Perfect signaling systems (PS) are coded red, perfect ambiguous systems (PA) are coded blue and non-perfect ambiguous systems (nPA) are coded darkgray. Experimental runs where participants failed to establish a joint communication protocol after 30 rounds are coded lightgray.

Symbol | Description |
---|---|

${t}_{i}\in T$ | information states of set T |

${s}_{i}\in S$ | signals of set S |

${r}_{i}\in R$ | response actions of set R |

${c}_{i}\in C$ | contextual cues of set C |

$Pr\in {(\Delta \left(T\right))}^{C}$ | probability function over T given $c\in C$ |

$U:T\times R\to \mathbb{R}$ | utility function |

$\sigma :T\to S$ | sender strategy |

$\rho :S\to R$ | receiver strategy (standard signaling game) |

$\rho :S\times C\to R$ | receiver strategy (context-signaling game) |

$\gamma =\langle \sigma ,\rho \rangle $ | communicative strategy (pair of sender + receiver strategy) |

LS Game | CS Game | CB Game | |
---|---|---|---|

number of states | 3 | 3 | 3 |

number of signals | 3 | 3 | 2 |

contextual cues | no | yes | yes |

LS Game | CB Game | CS Game | |
---|---|---|---|

number of sender strategies | 27 | 8 | 27 |

number of receiver strategies | 27 | 81 | 729 |

total number of strategies | 729 | 648 | 19,683 |

perfect signaling systems | 6 ($0.8\%$) | - | 54 ($0.27\%$) |

perfect ambiguous systems | - | 2 ($0.3\%$) | 54 ($0.27\%$) |

(non-perfect) evolutionarily stable sets | no | yes | yes |

LS Game | CB Game | CS Game | |
---|---|---|---|

perfect signaling system | $86\%$ | - | $34\%$ |

perfect ambiguous systems | - | $44\%$ | $35\%$ |

non-perfect ambiguous systems | $14\%$ | $56\%$ | $31\%$ |

Game | Recruitment | Participants | |
---|---|---|---|

Session I | LS game | Prolific | 10 ($5\times 2$) |

Session II | CS game | Invitation | 10 ($5\times 2$) |

Session III | CS game | Prolific | 10 ($5\times 2$) |

Session IV | CB game | Prolific | 10 ($5\times 2$) |

Session V | CB game | Prolific | 10 ($5\times 2$) |

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**MDPI and ACS Style**

Mühlenbernd, R.; Wacewicz, S.; Żywiczyński, P. The Evolution of Ambiguity in Sender—Receiver Signaling Games. *Games* **2022**, *13*, 20.
https://doi.org/10.3390/g13020020

**AMA Style**

Mühlenbernd R, Wacewicz S, Żywiczyński P. The Evolution of Ambiguity in Sender—Receiver Signaling Games. *Games*. 2022; 13(2):20.
https://doi.org/10.3390/g13020020

**Chicago/Turabian Style**

Mühlenbernd, Roland, Sławomir Wacewicz, and Przemysław Żywiczyński. 2022. "The Evolution of Ambiguity in Sender—Receiver Signaling Games" *Games* 13, no. 2: 20.
https://doi.org/10.3390/g13020020