# 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.

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**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