# Evolution of Decisions in Population Games with Sequentially Searching Individuals

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## Abstract

**:**

## 1. Introduction

## 2. The General Model

#### 2.1. Payoffs

#### 2.2. Decisions, Types and Strategies

#### 2.3. Expected Payoffs and Game Probabilities

#### 2.4. Replicator Equation

## 3. The Results

#### 3.1. (A) The Basic Model

**Figure 1.**Phase plane analysis for the basic model: Each panel presents a simplex where each point is a frequency distribution at some point in time with ${\sum}_{A}{p}_{A}=1$. With double-lines, we indicate the set of frequencies where each point is an equilibrium; with black color, we show which equilibrium is stable (not necessarily asymptotically stable) and with white color, which equilibria are unstable. Arrows indicate the direction of the dynamics along each eigenvector. Grey dashed arrows give an example of an invasion-substitution sequence. See Section 3.1 for the discussion of the evolutionary properties of this model and the Appendix for the full analytical treatment.

#### 3.1.1. Invasion of Choosy Decisions

#### 3.1.2. Best Strategies

#### 3.1.3. Evolution of Cooperation and Choosy Decisions

#### 3.2. (B) Imperfect Signals

**Figure 2.**Phase-plane analysis for the model with imperfect signals: see Figure 1 for explanations. In this slightly more complicated model, arrows are not drawn for the dimorphic equilibria (equilibria on the edges) that are not best strategy solutions or that do not affect the invasion substitution sequence. Trimorphic equilibria that are not the best strategy solutions or that do not affect the invasion-substitution sequence are not drawn at all. We found no four-morphic equilibria. Grey dashed arrows give an example of an invasion-substitution sequence. We only draw the sequences that are qualitatively different from the ones presented in Figure 1. See Section 3.2 for the discussion of the evolutionary properties of this model and the Appendix for further analysis.

#### 3.2.1. Invasion of Choosy Decisions

#### 3.2.2. Best Strategies

#### 3.2.3. Evolution of Cooperation and Choosy Decisions

#### 3.3. (C) Search Costs

**Figure 3.**Phase-plane analysis for the model with search costs: see Figure 1 for explanations. In this model, we draw separate phase planes only for qualitatively different best strategy solutions or invasion-substitution sequences and when the stability of the monomorphic equilibria changes. Similarly to Figure 2, arrows are not drawn for the dimorphic equilibria (equilibria on the edges) that are not best strategy solutions or that do not affect the invasion substitution sequence. Trimorphic equilibria that are not the best strategy solutions or that do not affect the invasion-substitution sequence are not drawn at all. We found no four-morphic equilibria. As before, we only draw the invasion-substitution sequences that are qualitatively different from the ones presented in previous figures. See Section 3.3 for the discussion of the evolutionary properties of this model and the Appendix for further analysis.

#### 3.3.1. Invasion of Choosy Decisions

#### 3.3.2. Best Strategies

#### 3.3.3. Evolution of Cooperation and Choosy Decisions

**(F)**$\frac{4}{3}(1-\frac{a}{b})<k<2$: The initial state ${D}_{1}$ is also an ESS and, thus, an endpoint of evolution.

#### 3.4. (D) The Full Model: Imperfect Signals and Search Costs

#### 3.4.1. Invasion of Choosy Decisions

#### 3.4.2. Best Strategies and the Evolution of Cooperation and Choosy Decisions

## 4. Discussion

**Figure 4.**Phase plane analysis for the full model: Similarly to Figure 3, in order to reduce the number of panels, we draw separate phase planes only for qualitatively different best strategy solutions or invasion-substitution sequences and when the stability of the monomorphic equilibria changes. In contrast to previous models, trimorphic and four-morphic ESS solutions were found, see (G) and (F), respectively. As before, we only draw the invasion-substitution sequences that are qualitatively different from the ones presented in previous figures. See Section 3.4 for the discussion of the evolutionary properties of this model and the Appendix for further analysis.

**Table 1.**A summary of the best strategy solutions that are reached by an invasion-substitution sequence starting from a non-choosy population. We collect all solutions for each model, and we group them according to the game that is played: snowdrift game ($0<k<1$) and a prisoner’s dilemma ($1<k<2$). Note that for each parameter value, there is only one evolutionary endpoint, but each model for each game may have many evolutionary endpoints. ESS, evolutionarily-stable strategy; BSS, best selective strategy.

Best Strategy Solutions (reached by evolution) | ||
---|---|---|

Snowdrift game ($0<k<1$) | Prisoner’s dilemma ($1<k<2$) | |

Non-choosy Population | ESS: $({C}_{1},{D}_{1})$ | ESS: ${D}_{1}$ |

Basic Model ($a=0,\epsilon =0$) | BSS: $({C}_{0},{C}_{1})$ | BSS: $({D}_{0},{D}_{1})$ |

Imperfect Signals ($a=0,\epsilon >0$) | ESS: $({C}_{0},{D}_{0})$ | BSS: $({D}_{0},{D}_{1})$ |

Search Costs ($a>0,\epsilon =0$) | ESS: $({C}_{1},{D}_{1})$ BSS: $({C}_{0},{C}_{1})$ | BSS: $({C}_{0},{C}_{1})$BSS: $({D}_{0},{D}_{1})$Evolutionary Cycle |

Full Model ($a>0,\epsilon >0$) | ESS: $({C}_{1},{D}_{1})$ESS: $({C}_{0},{D}_{0})$ESS: $({C}_{1},{C}_{0},{D}_{0})$ESS: $({C}_{1},{D}_{1},{C}_{0},{D}_{0})$Evolutionary Cycle | ESS: $({C}_{0},{D}_{0})$ESS: ${D}_{1}$Evolutionary Cycle |

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## A. Appendix: The Basic Model

#### A.1. The Existence and Stability of Equilibria

#### A.1.1. Monomorphic Equilibria

#### A.1.2. Dimorphic Equilibria

#### A.1.3. Trimorphic and Four-Morphic Equilibria

#### A.2. Invasion-Substitution Sequence

## B. Appendix: Imperfect Signals

#### B.1. The Existence and Stability of Equilibria

#### B.1.1. Monomorphic Equilibria

#### B.1.2. Dimorphic Equilibria

#### B.1.3. Trimorphic and Four-Morphic Equilibria

#### B.2. Invasion-Substitution Sequence

## C. Appendix: Search Costs

#### C.1. The Existence and Stability of Equilibria

#### C.1.1. Monomorphic Equilibria

#### C.1.2. Dimorphic Equilibria

#### C.1.3. Trimorphic and Four-Morphic Equilibria

#### C.2. Invasion-Substitution Sequence

## D. Appendix: Full Model with Imperfect Signals and Search Costs

## References

- Gale, D.; Shapley, L.S. College admissions and the stability of marriage. Am. Math. Mon.
**1962**, 69, 9–15. [Google Scholar] - Knuth, D.E. Mariages Stables et Leurs Relations Avec D’autres Problèmes Combinatoires; Les Presses de l’Université de Montréal: Montreal, QC, Canada, 1976. [Google Scholar]
- Roth, A.E. Incentive compatibility in a market with indivisible goods. Econ. Lett.
**1982**, 9, 127–132. [Google Scholar] [CrossRef] - Roth, A.E. The evolution of the labor market for medical interns and residents: A case study in game theory. J. Political Econ.
**1984**, 92, 991–1016. [Google Scholar] [CrossRef] - Roth, E.A.; Sotomayor, M.A.O. Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Kojima, F.; Pathak, P.A. Incentives and stability in large two-sided matching markets. Am. Econ. Rev.
**2009**, 99, 608–627. [Google Scholar] [CrossRef] - Ackermann, H.; Goldberg, P.W.; Mirrokni, V.S.; Röglin, H.; Vöcking, B. Uncoordinated two-sided matching markets. Siam J. Comput.
**2011**, 40, 92–106. [Google Scholar] [CrossRef] - Eriksson, K.; Häggström, O. Instability of matchings in decentralized markets with various preference structures. Int. J. Game Theory
**2008**, 36, 409–442. [Google Scholar] [CrossRef] - Kalick, S.M.; Hamilton, T.E. The matching hypothesis reexamined. J. Personal. Soc. Psychol.
**1986**, 51, 673–682. [Google Scholar] [CrossRef] - Johnstone, R.A. The tactics of mutual mate choice and competitive search. Behav. Ecol. Sociobiol
**1997**, 40, 51–59. [Google Scholar] [CrossRef] - Alpern, S.; Reyniers, D. Strategic mating with common preferences. J. Theor. Biol.
**2005**, 237, 337–354. [Google Scholar] [CrossRef] [PubMed] - Alpern, S.; Katranzi, I. Equilibria of two-sided matching games with common preferences. Eur. J. Oper. Res.
**2009**, 196, 1214–1222. [Google Scholar] [CrossRef] - Ramsey, D.M. Partnership formation based on multiple traits. Eur. J. Oper. Res.
**2012**, 216, 624–637. [Google Scholar] [CrossRef] - Alpern, S.; Katranzi, I.; Ramsey, D. Equilibrium population dynamics when mating is by mutual choice based on age. Theor. Poput. Biol.
**2014**, 94, 63–72. [Google Scholar] [CrossRef] [PubMed] - McNamara, J.M.; Collins, E.J. The job search problem as an employer-candidate game. J. Appl. Prob.
**1990**, 27, 815–827. [Google Scholar] [CrossRef] - Bloch, F.; Ryder, H. Two-sided search, marriages, and matchmakers. Internat. Econ. Rev.
**2000**, 41, 93–114. [Google Scholar] [CrossRef] - Shimer, R.; Smith, L. Assortative matching and search. Econometrica
**2000**, 68, 343–369. [Google Scholar] [CrossRef] - Eriksson, K.; Sjöstrand, J.; Strimling, P. Optimal expected rank in a two-sided secretary problem. Oper. Res.
**2007**, 55, 921–931. [Google Scholar] [CrossRef] - Ward, S.A. Optimal habitat selection in time-limited dispersers. Am. Nat.
**1987**, 129, 568–579. [Google Scholar] [CrossRef] - Real, L.A. Search theory and mate choice. I. Models of single-sex discrimination. Am. Nat.
**1990**, 136, 376–404. [Google Scholar] [CrossRef] - Backwell, P.R.Y.; Passmore, N.I. Time constraints and multiple choice criteria in the sampling behavior and mate choice of the fiddler crab, Uca annulipes. Behav. Ecol. Soc.
**1996**, 38, 407–416. [Google Scholar] [CrossRef] - Gray, D. Intrinsic factors affecting female choice in house crickets: time of first reproduction, female age, nutritional condition, body size, and size-relative reproductive investment. J. Insect Behav.
**1999**, 12, 691–700. [Google Scholar] - Tinghitella, R.M.; Weigel, E.G.; Head, M.; Boughman, J.W. Flexible mate choice when mates are rare and time is short. Ecol. Evol.
**2013**, 3, 2820–2831. [Google Scholar] [CrossRef] - Priklopil, T.; Kisdi, E.; Gyllenberg, M. Evolutionarily stable mating decisions for sequentially searching females and the stability of reproductive isolation by assortative mating. Evolution
**2015**, 69, 1015–1026. [Google Scholar] [CrossRef] [PubMed] - Nöe, R.; Hammerstein, P. Biological markets: Supply and demand determine the effect of partner choice in cooperation, mutualism and mating. Behav. Ecol. Sociobiol.
**1994**, 35, 1–11. [Google Scholar] [CrossRef] - Batali, J.; Kitcher, P. Evolution of altruism in optional and compulsory games. J. Theor. Biol.
**1995**, 175, 161–171. [Google Scholar] [CrossRef] [PubMed] - Sherratt, T.; Roberts, G. The Evolution of generosity and choosiness in cooperative exchanges. J. Theor. Biol.
**1998**, 193, 167–177. [Google Scholar] [CrossRef] - Hauert, C.; De Monte, S.; Hofbauer, J.; Sigmund, K. Volunteering as red queen mechanism for cooperation in public goods games. Science
**2002**, 296, 1129–1132. [Google Scholar] [CrossRef] [PubMed] - Hruschka, D.J.; Henrich, J. Friendship, cliquishness, and the emergence of cooperation. J. Theor. Biol.
**2006**, 239, 1–15. [Google Scholar] [CrossRef] [PubMed] - Fu, F.; Hauert, C.; Nowak, M.A.; Wang, L. Reputation-based partner choice promotes cooperation in social networks. Phys. Rev.
**2008**, E78, 026117. [Google Scholar] [CrossRef] - McNamara, J.M.; Barta, Z.; Fromhage, L.; Houston, A.I. The coevolution of choosiness and cooperation. Nature
**2008**, 451, 189–192. [Google Scholar] [CrossRef] [PubMed] - Ghang, W.; Nowak, M.A. Indirect reciprocity with optional interactions. J. Theor. Biol.
**2015**, 21, 1–11. [Google Scholar] [CrossRef] [PubMed] - Sigmund, K. Calculus of Selfishness; Princeton University Press: Princeton, NJ, USA, 2010. [Google Scholar]

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Priklopil, T.; Chatterjee, K.
Evolution of Decisions in Population Games with Sequentially Searching Individuals. *Games* **2015**, *6*, 413-437.
https://doi.org/10.3390/g6040413

**AMA Style**

Priklopil T, Chatterjee K.
Evolution of Decisions in Population Games with Sequentially Searching Individuals. *Games*. 2015; 6(4):413-437.
https://doi.org/10.3390/g6040413

**Chicago/Turabian Style**

Priklopil, Tadeas, and Krishnendu Chatterjee.
2015. "Evolution of Decisions in Population Games with Sequentially Searching Individuals" *Games* 6, no. 4: 413-437.
https://doi.org/10.3390/g6040413