# Does Informational Equivalence Preserve Strategic Behavior? Experimental Results on Trockel’s Model of Selten’s Chain Store Story

## Abstract

**:**

## 1. Introduction

#### Experimental Literature

## 2. Experimental Design

#### 2.1. Subjects

#### 2.2. Types

#### 2.3. In-Experiment Matchings

#### 2.4. Periods

#### 2.5. History Tables

#### 2.6. Treatments

**First treatment T1:**Trockel’s game is used with the original payoffs.**Second treatment T2:**It differs from T1 in the payoff of the monopoly when the outcome is ‘Aggressive-Out’. In T2, the payoff increased from 5 to 10 when the entrant chose ‘Out’, to enhance the appeal of ‘Aggressive’ play.

#### 2.7. Questionnaire and the Payment

## 3. Results

#### 3.1. Logistic Regression Analysis

#### 3.2. Questionnaire Data

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Participant Instructions

#### Appendix A.1.1. Types of Participants

#### Appendix A.1.2. In-experiment Matchings

#### Appendix A.1.3. Periods

A’s Point(s) | B’s Point(s) | |
---|---|---|

A’s Decision is ‘Z’ & B’s Decision is ‘X’ | 0 | 0 |

A’s Decision is ‘Z’ & B’s Decision is ‘Y’ | 5 | 1 |

A’s Decision is ‘T’ & B’s Decision is ‘X’ | 2 | 2 |

A’s Decision is ‘T’ & B’s Decision is ‘Y’ | 5 | 1 |

- If the decision of type A participant is Z and the decision of type B participant is X, the points of the type A participant decrease by 1 point. Her earnings in this round are saved as 0 point.
- If the decision of type A participant is Z and the decision of type B participant is Y, the points of the type A participant increase by 4 points. Her earnings in this round are saved as 5 points.
- If the decision of type A participant is T and the decision of type B participant is X, the points of the type A participant increase by 1 point. Her earnings in this round are saved as 2 points.
- If the decision of type A participant is T and the decision of type B participant is Y, the points of the type A participant increase by 4 points. Her earnings in this round are saved as 5 points.
- If the decision of type A participant is Z and the decision of type B participant is X, the points of the type B participant decrease by 1 point. Her earnings in this round are saved as 0 point.
- If the decision of type A participant is Z and the decision of type B participant is Y, the points of the type B participant remain constant. Her earnings in this round are saved as 1 point.
- If the decision of type A participant is T and the decision of type B participant is X, the points of the type B participant increase by 1 point. Her earnings in this round are saved as 2 points.
- If the decision of type A participant is T and the decision of type B participant is Y, the points of the type B participant remain constant. Her earnings in this round are saved as 1 point.

My Points | 0 | A’s Points | 0 |

B’s Points | 0 | My Points | 0 |

Round | My Decision | My Opponent’s Decision | My Points |

1 | M | P | 9 |

2 | ? | ? | ? |

Round | The Decision of B Who Was Active in That Round | The Points of B Who Was Active in That Round | A’s Point |

1 | P | 9 | 9 |

2 | ? | ? | ? |

#### Appendix A.1.4. In-experiment Earnings

#### Appendix A.2. Questionnaire

#### Appendix A.2.1. Type A Questionnaire 1

- (A)
- I tried to earn as much as I can. I played by trying to maximize my earnings.
- (B)
- I tried to be fair towards type B participants. For me, the total of the earnings of all participants was also important, I did not consider only myself.
- (C)
- As a type A participant, the most important thing was my reputation. I tried to be a ‘tough’ player, not to make type B participants benefit.
- (D)
- My decisions depend on other reasons.

#### Appendix A.2.2. Type B Questionnaire

- (A)
- Whomever I faced, my strategy was to always play ‘X’ or always ‘Y’.
- (B)
- I made my choices between ‘X’ and ‘Y’ considering the number of the round in which I decided.
- (C)
- My decisions depend on other reasons.
- (D)
- I made my choices between ‘X’ and ‘Y’ considering the ideas and the beliefs about the type A participant I faced. I examined how she played in comparison to the other type B participants before me.

#### Appendix A.2.3. Type A Questionnaire 2

- (A)
- In each round I decided independently from the other rounds. I did not think that my decision to a type B participant could affect the other type B participants’ decisions.
- (B)
- I thought that my decision to a type B participant could affect the following type B participant’s decision and only this participant.
- (C)
- I thought that my decision to a type B participant could affect all of the following type B participants’ decisions.
- (D)
- I thought that my decision to a type B participant could affect all the following type B participants’ decisions, except a few of them from the last round.

## References

- Selten, R. The chain store paradox. Theory Decis.
**1978**, 9, 127–159. [Google Scholar] [CrossRef] - Kreps, D.M.; Wilson, R. Reputation and imperfect information. J. Econ. Theory
**1982**, 27, 253–279. [Google Scholar] [CrossRef] [Green Version] - Milgrom, P.; Roberts, J. Predation, reputation, and entry deterrence. J. Econ. Theory
**1982**, 27, 280–312. [Google Scholar] [CrossRef] [Green Version] - Trockel, W. The chain-store paradox revisited. Theory Decis.
**1986**, 21, 163–179. [Google Scholar] [CrossRef] [Green Version] - Dalkey, N. Equivalence of information patterns and essentially determinate games. In Contributions to the Theory of Games; Kuhn, H.W., Tucker, A.W., Eds.; Princeton University Press: Princeton, NJ, USA, 1953; Volume 2, pp. 217–244. [Google Scholar]
- Thompson, F.A. Equivalence of games in extensive form. In Classics in Game Theory; Kuhn, H.W., Ed.; Princeton University Press: Princeton, NJ, USA, 1997; pp. 36–45. [Google Scholar]
- Malinvaud, E. Note on von Neumann-Morgenstern’s strong independence axiom. Econom. J. Econom. Soc.
**1952**, 20, 679. [Google Scholar] [CrossRef] - Sundali, J.A.; Rapoport, A. Induction vs. Deterrence in the Chain Store Game: How Many Potential Entrants are Needed to Deter Entry? In Understanding Strategic Interaction; Albers, W., Güth, W., Hammerstein, P., Moldovanu, B., van Damme, E., Eds.; Springer: Berlin/Heidelberg, Germany, 1997; pp. 403–417. [Google Scholar]
- Posner, R.A. Economic Analysis of Law; Wolters Kluwer Law & Business: New York, NY, USA, 2014. [Google Scholar]
- Gates, S.; Humes, B.D. Games, Information, and Politics: Applying Game Theoretic Models to Political Science; University of Michigan Press: Ann Arbor, MI, USA, 1997. [Google Scholar]
- Camerer, C.; Weigelt, K. Experimental tests of a sequential equilibrium reputation model. Econom. J. Econom. Soc.
**1988**, 56, 1–36. [Google Scholar] [CrossRef] [Green Version] - Jung, Y.J.; Kagel, J.H.; Levin, D. On the existence of predatory pricing: An experimental study of reputation and entry deterrence in the chain-store game. RAND J. Econ.
**1994**, 25, 72–93. [Google Scholar] [CrossRef] - Sundali, J.; Israeli, A.; Janicki, T. Reputation and deterrence: Experimental evidence from the Chain Store Game. J. Bus. Econ. Stud.
**2000**, 6, 1. [Google Scholar] - Benard, S. Reputation systems, aggression, and deterrence in social interaction. Soc. Sci. Res.
**2013**, 42, 230–245. [Google Scholar] [CrossRef] [PubMed] - Neugebauer, T. Airness and reciprocity in the hawk–dove game. J. Econ. Behav. Organ.
**2008**, 66, 243–250. [Google Scholar] [CrossRef] [Green Version] - Sundali, J. An Experimental Investigation of Market Entry Problems. Ph.D. Thesis, The University of Arizona, Tucson, AZ, USA, 1995. [Google Scholar]
- Kahneman, D.; Tversky, A. Prospect theory: An analysis of decision under risk. Econom. J. Econom. Soc.
**1979**, 47, 263–291. [Google Scholar] [CrossRef] [Green Version] - Tversky, A.; Kahneman, D. The framing of decisions and the psychology of choice. Science
**1981**, 211, 453–458. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rapoport, A. Order of play in strategically equivalent games in extensive form. Int. J. Game Theory
**1997**, 26, 113–136. [Google Scholar] [CrossRef] - Güth, W.; Huck, S.; Rapoport, A. The limitations of the positional order effect: Can it support silent threats and non-equilibrium behavior? J. Econ. Behav. Organ.
**1998**, 34, 313–325. [Google Scholar] [CrossRef] - Fischbacher, U. z-Tree: Zurich toolbox for ready-made economic experiments. Exp. Econ.
**2007**, 10, 171–178. [Google Scholar] [CrossRef] [Green Version] - Greiner, B. The Online Recruitment System ORSEE 2.0. A Guide for the Organization of Experiments in Economics. Work. Pap. Ser. Econ.
**2004**, 10, 63–104. [Google Scholar] - Selten, R.; Stoecker, R. End behavior in sequences of finite Prisoner’s Dilemma supergames: A learning theory approach. J. Econ. Behav. Organ.
**1986**, 7, 47–70. [Google Scholar] [CrossRef] - Saaty, T.L. The Analytic Hierarchy Process; McGraw Hill: New York, NY, USA, 1980. [Google Scholar]
- Kreps, D.M. Game Theory and Economic Modelling; Oxford University Press: Oxford, UK, 1990. [Google Scholar]

1. | One would apply transformations preserving the informational structure of a perfect information game: inflating the game and reverting the order of moves, both under preserving each player’s information regarding other player’s and one’s own past moves. |

2. | If the payoff of a monopoly player is 1, the entrant of that round plays ‘Out’. Hence, this payoff does not contain any information related to the decision of the monopoly player. |

Round | My Decision | My Opponent’s Decision | My Points |
---|---|---|---|

1 | Coop | Out | 5 |

2 | Agg | In | 0 |

3 | ? | ? | ? |

Round | The Decision of Type E who was Active in that Round | The Points of Type E who was Active in that Round | The Points of the Opponent |
---|---|---|---|

1 | Out | 1 | 5 |

2 | In | 0 | 0 |

3 | ? | ? | ? |

Round | Agg. Frequency | P(Agg) | In Frequency | P(In) |
---|---|---|---|---|

1 | 17 | 18% | 65 | 68% |

2 | 35 | 36% | 62 | 65% |

3 | 22 | 23% | 64 | 67% |

4 | 33 | 34% | 66 | 69% |

5 | 32 | 33% | 68 | 71% |

6 | 32 | 33% | 72 | 75% |

7 | 34 | 35% | 69 | 72% |

8 | 25 | 26% | 78 | 81% |

9 | 27 | 28% | 74 | 77% |

10 | 21 | 22% | 84 | 88% |

11 | 32 | 33% | 77 | 80% |

12 | 31 | 32% | 70 | 73% |

13 | 32 | 33% | 81 | 84% |

14 | 34 | 35% | 74 | 77% |

15 | 25 | 26% | 82 | 85% |

16 | 28 | 29% | 70 | 73% |

17 | 26 | 27% | 79 | 82% |

18 | 22 | 23% | 73 | 76% |

19 | 20 | 21% | 77 | 80% |

20 | 16 | 17% | 82 | 85% |

Total | 544 | 1467 | ||

Mean | 27.20 | 0.29 | 73.35 | 0.76 |

Std. Dev. | 5.98 | 0.07 | 6.55 | 0.07 |

Action | Odd Ratio | Std.Err. | z | $\mathit{P}>\mid \mathit{z}\mid $ | $\%95$ Conf. Interval | |
---|---|---|---|---|---|---|

Agg | 0.7868992 | 0.0330637 | −5.70 | 0.000 *** | 0.7246922 | 0.8544459 |

0.7723197 | 0.0306647 | −6.51 | 0.000 *** | 0.714497 | 0.8348219 | |

Uncertain | 0.813862 | 0.0371601 | −4.51 | 0.000 *** | 0.7441934 | 0.8900528 |

0.8244335 | 0.0355853 | −4.47 | 0.000 *** | 0.7575564 | 0.8972145 | |

Round | 1.157306 | 0.0247897 | 6.82 | 0.000 *** | 1.109725 | 1.206927 |

1.142695 | 0.022985 | 6.63 | 0.000 *** | 1.098522 | 1.188645 | |

PRP | 1.223414 | 0.1132568 | 2.18 | 0.029 * | 1.020409 | 1.466807 |

1.423831 | 0.1165063 | 4.32 | 0.000 *** | 1.212853 | 1.67151 | |

Uni | 1 | (omitted) | ||||

0.7344255 | 0.1119438 | −2.03 | 0.043 * | 0.5447591 | 0.9901271 | |

Faculty | 1 | (omitted) | ||||

0.9905217 | 0.0520703 | −0.18 | 0.856 | 0.8935472 | 1.098021 | |

Gender | 1 | (omitted) | ||||

0.6538374 | 0.0976447 | −2.85 | 0.004 *** | 0.4879222 | 0.8761711 | |

Game Theory | 1 | (omitted) | ||||

1.384987 | 0.213229 | 2.12 | 0.034 * | 1.024229 | 1.872814 | |

Constant | ||||||

2.597589 | 0.6460961 | 3.84 | 0.000 | 1.595335 | 4.229501 |

Action | Odd Ratio | Std.Err. | z | $\mathit{P}>\mid \mathit{z}\mid $ | $\%95$ Conf. Interval | |
---|---|---|---|---|---|---|

PRP | 1.368848 | 0.1043251 | 4.12 | 0.000 *** | 1.178914 | 1.589383 |

1.517539 | 0.1053483 | 6.01 | 0.000 *** | 1.324491 | 1.738724 | |

LRA | 0.4019908 | 0.0698842 | −5.24 | 0.000 *** | 0.2859162 | 0.5651888 |

0.4431788 | 0.0690054 | −5.23 | 0.000 *** | 0.3266194 | 0.6013341 | |

LRU | 0.5323112 | 0.0885989 | −3.79 | 0.000 *** | 0.3841404 | 0.7376346 |

0.4844501 | 0.0727617 | −4.83 | 0.000 *** | 0.3609136 | 0.6502717 | |

Uni | 1 | (omitted) | ||||

0.7629754 | 0.1124712 | −1.84 | 0.066 | 0.5715231 | 1.018561 | |

Faculty | 1 | (omitted) | ||||

0.9960464 | 0.0511204 | −0.08 | 0.938 | 0.9007268 | 1.101453 | |

Gender | 1 | (omitted) | ||||

0.6699686 | 0.097497 | −2.75 | 0.006 *** | 0.5037135 | 0.8910977 | |

Game Theory | 1 | (omitted) | ||||

1.361758 | 0.2046238 | 2.05 | 0.04 * | 1.014365 | 1.828122 | |

Constant | ||||||

4.732442 | 1.144424 | 6.43 | 0.000 | 2.946071 | 7.601992 |

Action | Odd Ratio | Std.Err. | z | $\mathit{P}>\mid \mathit{z}\mid $ | $\%95$ Conf. Interval | |
---|---|---|---|---|---|---|

Agg-In | 0.7811345 | 0.042526 | −4.54 | 0.000 *** | 0.7020778 | 0.8690933 |

1.00305 | 0.0528328 | 0.06 | 0.954 | 0.9046656 | 1.112134 | |

Agg-Out | 1.495115 | 0.1235306 | 4.87 | 0.000 *** | 1.271586 | 1.757937 |

1.608184 | 0.112841 | 6.77 | 0.000 *** | 1.401554 | 1.845278 | |

Coop-In | 1.007013 | 0.0260432 | 0.27 | 0.787 | 0.9572416 | 1.059373 |

0.8806632 | 0.0221948 | −5.04 | 0.000 *** | 0.8382191 | 0.9252565 | |

Uni | 1 | (omitted) | ||||

0.8516906 | 0.1717452 | −0.80 | 0.426 | 0.5736344 | 1.264528 | |

Faculty | 1 | (omitted) | ||||

0.9682641 | 0.0615046 | −0.51 | 0.612 | 0.8549193 | 1.096636 | |

Gender | 1 | (omitted) | ||||

0.9346882 | 0.1849402 | −0.34 | 0.733 | 0.6342281 | 1.377489 | |

Game Theory | 1 | (omitted) | ||||

1.320798 | 0.2717673 | 1.35 | 0.176 | 0.8824567 | 1.976877 | |

Constant | ||||||

0.4324311 | 0.1158132 | −3.13 | 0.002 | 0.2558286 | 0.730945 |

Action | Odd Ratio | Std.Err. | z | $\mathit{P}>\mid \mathit{z}\mid $ | $\%95$ Conf. Interval | |
---|---|---|---|---|---|---|

LRI | 1.068754 | 0.1542534 | 0.46 | 0.645 | 0.8054228 | 1.418182 |

0.9924131 | 0.1425513 | −0.05 | 0.958 | 0.748902 | 1.315104 | |

Uni | 1 | (omitted) | ||||

1.084298 | 0.3698836 | 0.24 | 0.812 | 0.5556231 | 2.116005 | |

Faculty | 1 | (omitted) | ||||

0.9248028 | 0.1001964 | −0.72 | 0.471 | 0.7478715 | 1.143592 | |

Gender | 1 | (omitted) | ||||

1.036639 | 0.3481785 | 0.11 | 0.915 | 0.5367013 | 2.002271 | |

Game Theory | 1 | (omitted) | ||||

1.230763 | 0.4302212 | 0.59 | 0.553 | 0.6203426 | 2.441842 | |

Constant | ||||||

0.3301953 | 0.1476151 | −2.48 | 0.013 | 0.1374795 | 0.7930558 |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Duman, P.
Does Informational Equivalence Preserve Strategic Behavior? Experimental Results on Trockel’s Model of Selten’s Chain Store Story. *Games* **2020**, *11*, 9.
https://doi.org/10.3390/g11010009

**AMA Style**

Duman P.
Does Informational Equivalence Preserve Strategic Behavior? Experimental Results on Trockel’s Model of Selten’s Chain Store Story. *Games*. 2020; 11(1):9.
https://doi.org/10.3390/g11010009

**Chicago/Turabian Style**

Duman, Papatya.
2020. "Does Informational Equivalence Preserve Strategic Behavior? Experimental Results on Trockel’s Model of Selten’s Chain Store Story" *Games* 11, no. 1: 9.
https://doi.org/10.3390/g11010009