# When Efficiency Requires Arbitrary Discrimination: Theoretical and Experimental Analysis of Equilibrium Selection

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- Is the third party more likely to choose an asymmetric sanctioning schedule when the two agents differ in wealth than when they are equally wealthy?
- Is the third party more likely to choose an asymmetric sanctioning schedule when the two agents can observe the third agent’s choice than when not?
- Does an asymmetric sanctioning schedule induce the two agents to coordinate their decisions to achieve an efficiency-enhancing outcome?

## 2. Theory

#### 2.1. The Sequential Order Protocol

**Proposition**

**1.**

- At the first decision stage, X chooses $p=0$ with probability w and $p=d$ with probability $1-w$ such that $w\in [0,1]$;
- At the second decision stage,
- –
- if $p\in \{0,d\}$, then there are two pure-strategy equilibria, $({s}_{Y},{s}_{Z})=(R,B)$ and $({s}_{Y},{s}_{Z})=(B,R)$, only one of which is strict;
- –
- if $p\in (0,d)$, then there also exists a mixed-strategy equilibrium $({\sigma}_{Y},{\sigma}_{Z})$, in addition to the two pure-strategy equilibria.

**Proposition**

**2.**

- At the first decision stage, X chooses $p=0$ and $p=d$ with equal probability;
- At the second decision stage,
- –
- if $0\le p<\frac{d}{2}$, $({s}_{Y},{s}_{Z})=(R,B)$;
- –
- if $\frac{d}{2}<p\le 1$, $({s}_{Y},{s}_{Z})=(B,R)$;
- –
- if $p=\frac{d}{2}$, $({\sigma}_{Y},{\sigma}_{Z})$ presented in (1).

#### 2.2. The Simultaneous Order Protocol

**Proposition**

**3.**

**Proposition**

**4.**

## 3. Experimental Design

#### 3.1. Treatments and Hypotheses

**Hypothesis**

**1.**

**Hypothesis**

**2.**

**Hypothesis**

**3.**

#### 3.2. Procedure

## 4. Results

#### 4.1. X’s Behavior

**Result**

**1.**

#### 4.2. Behavior of Y and Z

**Result**

**2.**

**Result**

**3.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Instructions for Treatment Seq-Asym (Originally Written in German)

#### Appendix A.1. Introduction

#### Appendix A.2. Detailed Information on the Experiment

#### Appendix A.3. Description of the Task

Combinations | |||||||||||

Black | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

White | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |

#### Appendix A.4. How to Compute Payoffs

- 1.
- If both Y and Z chose Red in Stage 2,$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill \mathrm{X}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =15\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \\ \hfill \mathrm{Y}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =15\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \\ \hfill \mathrm{Z}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =15\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \end{array}\\ \hline\end{array}$$
- 2.
- If Y chose Red and Z chooses Blue in Stage 2,$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill \mathrm{X}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =15\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \\ \hfill \mathrm{Y}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =15\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \\ \hfill \mathrm{Z}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =30\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \end{array}\\ \hline\end{array}$$
- 3.
- If Y chose Blue and Z chooses Red in Stage 2,$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill \mathrm{X}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =15\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \\ \hfill \mathrm{Y}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =30\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \\ \hfill \mathrm{Z}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =15\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \end{array}\\ \hline\end{array}$$
- 4.
- If both Y and Z chose Blue in Stage 2,$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill \mathrm{X}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =10\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \\ \hfill \mathrm{Y}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =15-\mathrm{number}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{black}\phantom{\rule{4.pt}{0ex}}\mathrm{ball}\left(\mathrm{s}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{combination}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \\ \hfill \mathrm{Z}\u2019\mathrm{s}\phantom{\rule{4.pt}{0ex}}\mathrm{payoff}& =15-\mathrm{number}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{white}\phantom{\rule{4.pt}{0ex}}\mathrm{ball}\left(\mathrm{s}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{combination}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{points}\hfill \end{array}\\ \hline\end{array}$$

#### Appendix A.5. Feedback Information at the End of Each Round

- Your decision as well as the decisions of the other two members in your group;
- Your payoff for the current round.

#### Appendix A.6. End of the Experiment

- The round chosen by the volunteer;
- The points you have earned in the chosen round;
- The corresponding earnings in euros;
- Your show-up fee;
- Your total earnings in euros.

- Please remain at your cubicle until asked to come forward and receive payment for the experiment.

## Appendix B. Control Questions for Treatment Seq-Asym (Originally Written in German)

- Q.1 (True/False Question) Your role will change every round.$$\mathrm{True}/\mathrm{False}$$
- Answer: False. Your role will stay the same throughout the experiment.

- Q.2 (True/False Question) You may be matched with the same participants more than once.$$\mathrm{True}/\mathrm{False}$$
- Answer: False. You will never be matched with the same participants again.

- Q.3 (True/False Question) Each participant will be paid a EUR 2.50 show-up fee.$$\mathrm{True}/\mathrm{False}$$
- Answer: False. The show-up fee differs across the three roles. You will receive a EUR 5.00 show-up fee if you are X, a EUR 2.50 show-up fee if you are Y, and a EUR 7.50 show-up fee if you are Z.

- Q.4 (Multiple Choice Question) In Stage 1, X chose a combination with 7 black balls and 3 white balls. In Stage 2, both Y and Z chose Red. How many points will they earn for this round?
- A.
- X will earn 10 points, Y will earn 8 points, and Z will earn 12 points;
- B.
- X will earn 10 points, Y will earn 12 points, and Z will earn 8 points;
- C.
- X will earn 15 points, Y will earn 30 points, and Z will earn 15 points;
- D.
- None of the above.

- Answer: D. Since both Y and Z chose Red in Stage 2, each of the group members will earn 15 points.

- Q.5 (Multiple Choice Question) In Stage 1, X chose a combination with 4 black balls and 6 white balls. In Stage 2, both Y and Z chose Blue. How many points will they earn for this round?
- A.
- X will earn 10 points, Y will earn 9 points, and Z will earn 11 points;
- B.
- X will earn 10 points, Y will earn 11 points, and Z will earn 9 points;
- C.
- Each of them will earn 15 points;
- D.
- None of the above.

- Answer: B. In Stage 1, X chose a combination with 4 black balls and 6 white balls. Since both Y and Z chose Blue in Stage 2, X will earn 10 points, Y will earn 11 ($=15-4$) points, Z will earn 9 ($=15-6$) points.

- Q.6 (Multiple Choice Question) In Stage 1, X chose a combination with 1 black ball and 9 white balls. In Stage 2, Y chose Red and Z chose Blue. How many points will they earn for this round?
- A.
- X will earn 10 points, Y will earn 6 points, and Z will earn 14 points;
- B.
- X will earn 10 points, Y will earn 14 points, and Z will earn 6 points;
- C.
- X will earn 15 points, Y will earn 15 points, and Z will earn 30 points;
- D.
- None of the above.

- Answer: C. Since Y chose Red and Z chose Blue in Stage 2, both X and Y will earn 15 points each, whereas Z will earn 30 points.

## Notes

1 | Arbitrarily selecting only one polluter as fully responsible parallels the legal practice of holding just one out of multiple culprits accountable for the entire damage. |

2 | In best-shot public good games where the level of public good provision is contingent on the largest individual contribution, efficiency requires that only one of several contributors should contribute the efficient level. |

3 | When attributing responsibility, judges are often influenced by case-unrelated discrepancies. For instance, they may hold richer employers responsible in labor disputes based on their financial status. |

4 | Roth and Malouf [1] studied equilibrium selection in bargaining scenarios. |

5 | In the ultimatum game, it is an equilibrium in weakly dominated strategies when the proposer offers more than the minimal amount which the responder would accept. |

6 | Feldhaus and Stauf [10] theoretically and experimentally examined whether cheap talk could lead to an efficient outcome in a three-person VD game. In the second treatment of their experiment, one randomly chosen player had to send the other group members a one-way message regarding the strategy she would play. In sharp contrast to cheap talk in their game, the third party’s sanctioning schedule is a costless but binding decision. |

7 | The assumption that X is not at all affected by the p-choice can be justified by viewing X as an independent entity, thus eliminating confounding efficiency concerns associated with varying total punishment levels. |

8 | Specifically interesting scenarios of such a game class are small c and large d, cases where X does not suffer much but is free in allocating sanctions, as well as large c rendering X as seriously harmed and strongly motivated to avoid common polluting. Additionally, scenarios with small b and large d could discourage Y and B from choosing different strategies. |

9 | Isomorphic invariance implies symmetry invariance. |

10 | Since we could not recruit twenty-seven participants for the second session of treatment Sim-Sym, we decided to run this session with eighteen participants instead. |

11 | The English instructions and control questions for treatment Seq-Asym are available in Appendix A and Appendix B, respectively. |

12 | An additional avenue for exploration could have been to investigate how advice suggesting that (extreme) discrimination is more likely to enhance efficiency would impact game playing, particularly regarding the sanctioning behavior of X participants. |

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**Figure 1.**Choice frequencies of Y and Z participants under the sequential order protocol with the locally estimated scatterplot smoother.

**Table 1.**The trimatrix subgame after X’s choice of $p\in [0,d]$ with players Y and Z and payoffs listed in alphabetic order.

Z | R | B | |

Y | |||

R | $e,e,e$ | $e,e,e+b$ | |

B | $e,e+b,e$ | $e-c,e-p,e-(d-p)$ |

Z | R | B | |

Y | |||

R | $15,15,15$ | $15,15,30$ | |

B | $15,30,15$ | $10,15-p,5+p$ |

Treatment | Round | Number of Black Balls | Mean | S.D. | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||||

Sim-Sym | 1 | 0 | 0 | 0 | 0 | 1 | 10 | 3 | 0 | 0 | 0 | 1 | 5.467 | 1.356 |

2 | 0 | 0 | 1 | 1 | 2 | 8 | 1 | 0 | 0 | 1 | 1 | 5.2 | 2.007 | |

3 | 0 | 0 | 0 | 0 | 0 | 11 | 1 | 1 | 1 | 0 | 1 | 5.733 | 1.486 | |

Seq-Sym | 1 | 2 | 1 | 0 | 0 | 1 | 9 | 0 | 2 | 2 | 0 | 1 | 5 | 2.635 |

2 | 3 | 1 | 1 | 0 | 1 | 8 | 1 | 0 | 0 | 1 | 2 | 4.556 | 3.110 | |

3 | 3 | 1 | 1 | 0 | 0 | 8 | 1 | 0 | 0 | 1 | 3 | 4.889 | 3.359 | |

Sim-Asym | 1 | 3 | 1 | 1 | 3 | 0 | 6 | 2 | 1 | 1 | 0 | 0 | 3.833 | 2.455 |

2 | 3 | 1 | 2 | 3 | 2 | 4 | 0 | 0 | 0 | 1 | 2 | 3.944 | 3.152 | |

3 | 5 | 1 | 0 | 1 | 2 | 6 | 0 | 0 | 1 | 1 | 1 | 3.833 | 3.185 | |

Seq-Asym | 1 | 2 | 2 | 1 | 5 | 4 | 2 | 0 | 0 | 0 | 1 | 1 | 3.556 | 2.640 |

2 | 4 | 1 | 1 | 1 | 2 | 3 | 1 | 2 | 1 | 1 | 1 | 4.222 | 3.246 | |

3 | 5 | 1 | 1 | 3 | 1 | 2 | 1 | 0 | 1 | 2 | 1 | 3.778 | 3.457 |

Model 1 | Model 2 | |
---|---|---|

Asym | −1.939 ** | −0.768 ** |

(0.901) | (0.313) | |

Seq | −1.058 | −0.320 |

(0.879) | (0.226) | |

Asym × Seq | −0.307 | 0.287 |

(1.171) | (0.394) | |

Constant | 0.870 | |

(0.665) | ||

N | 207 | 207 |

Role | Round | Sim-Sym | Sim-Asym | Seq-Sym | Seq-Asym | ||||
---|---|---|---|---|---|---|---|---|---|

Red | Blue | Red | Blue | Red | Blue | Red | Blue | ||

Y | 1 | 8 | 7 | 7 | 11 | 7 | 11 | 4 | 14 |

2 | 7 | 8 | 7 | 11 | 6 | 12 | 8 | 10 | |

3 | 4 | 11 | 9 | 9 | 5 | 13 | 7 | 11 | |

Z | 1 | 8 | 7 | 9 | 9 | 10 | 8 | 13 | 5 |

2 | 4 | 11 | 9 | 9 | 7 | 11 | 8 | 10 | |

3 | 3 | 12 | 6 | 12 | 7 | 11 | 9 | 9 |

Model 3 | Model 4 | Model 5 | |
---|---|---|---|

Asym | −0.177 | −0.466 | −0.091 |

(0.312) | (0.292) | (0.243) | |

Seq | −0.049 | ||

(0.266) | |||

Asym × Seq | −0.022 | ||

(0.372) | |||

NumBB | −0.333 *** | 0.227 *** | |

(0.123) | (0.070) | ||

Constant | 0.384 * | 2.218 *** | −0.936 ** |

(0.216) | (0.763) | (0.370) | |

N | 414 | 108 | 108 |

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## Share and Cite

**MDPI and ACS Style**

Güth, W.; Otsubo, H.
When Efficiency Requires Arbitrary Discrimination: Theoretical and Experimental Analysis of Equilibrium Selection. *Games* **2023**, *14*, 65.
https://doi.org/10.3390/g14050065

**AMA Style**

Güth W, Otsubo H.
When Efficiency Requires Arbitrary Discrimination: Theoretical and Experimental Analysis of Equilibrium Selection. *Games*. 2023; 14(5):65.
https://doi.org/10.3390/g14050065

**Chicago/Turabian Style**

Güth, Werner, and Hironori Otsubo.
2023. "When Efficiency Requires Arbitrary Discrimination: Theoretical and Experimental Analysis of Equilibrium Selection" *Games* 14, no. 5: 65.
https://doi.org/10.3390/g14050065