# Choices in the 11–20 Game: The Role of Risk Aversion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}× 100, and so on. The unique equilibrium is to guess 0.

## 2. Related Literature

## 3. Theoretical Analysis

^{2}× 100, and so on. The unique equilibrium is to choose 0 for every player.

^{1−r}/(1 − r), where r is the degree of (relative) risk aversion and x is the amount of money obtained. When r=1.37, the predicted percentages are as follows: 20 (8%), 19 (16%), 18 (23.3%), 17 (30.5%), 16 (22.2%), and 15 and below (0%). We can also compute the mixed strategy equilibrium for the case where players have constant absolute risk aversion (CARA) utilities; i.e., players have utility functions u(x) = 1 ‒ e

^{−αx}, where α is the degree of (absolute) risk aversion. When α = 0.15, the predicted percentages are as follows: 20 (14.7%), 19 (27.2%), 18 (38.1%), 17 (20%), and 16 and below (0%). Evidently, as we can see from Table 1, the prediction of the choice of levels 1, 2, and 3 are closer to the data in Arad and Rubinstein (2012) [3] and our experiment than the prediction based on the equilibrium of risk neutral players.

^{1−r}/(1 − r), where each player’s risk aversion level r is the player’s private information. Assume that each player’s r is drawn from a uniform distribution on [0, 2], which is common knowledge. It can be verified that the equilibrium is that the player with 1.815 < r < 2 will choose 20; the player with 1.4708 < r < 1.815 will choose 19; the player with 0.9933 < r < 1.4708 will choose 18; the player with 0.4051 < r < 0.9933 will choose 17; and the player with 0 < r < 0.4051 will choose 16 (see the Appendix for a proof). Thus, the more risk-averse the player is, the more likely the higher number will be chosen.2 The choice probabilities of 20, 19, 18, 17, and 16 are roughly 9%, 17%, 24%, 29.5%, and 20.5%, respectively.

## 4. Experimental Design

## 5. Experimental Results

#### 5.1. Measurement of the Depth of Thinking

#### 5.2. Risk aversion and choice in the 11–20 Game

^{5}. 5 It is found that the depth of thinking is negatively correlated with risk premium. The result also suggests that there is a non-linear effect on the relationship between risk premium and depth of thinking in the 11–20 game. In particular, the higher is the risk aversion, the larger is the marginal effect.

#### 5.3. The relationship between choices in the P beauty contest game and the 11–20 game

#### 5.4. Comparison of results in P Beauty to Nagel (1995)

#### 5.5. Comparison of results in 11–20 to Arad and Rubinstein (2012)

## 6. Discussion

## Appendix A

**Figure A1.**Depth of Thinking and Risk Aversion. (

**A**) Depth of Thinking in the P-beauty Contest Game and Risk Aversion; (

**B**) Depth of Thinking in the 11–20 Game and Risk Aversion.

Choice | Gamble A | Gamble B | Expected Value of Gamble A | Expected Value of Gamble B |
---|---|---|---|---|

1 | 1/10 , RMB 20 ; 9/10, RMB 16 | 1/10 , RMB 38.5 ; 9/10, RMB 1 | 16.4 | 4.75 |

2 | 2/10 , RMB20 ; 8/10, RMB16 | 2/10 , RMB 38.5 ; 8/10, RMB 1 | 16.8 | 8.5 |

3 | 3/10 , RMB20 ; 7/10, RMB16 | 3/10 , RMB 38.5 ; 7/10, RMB 1 | 17.2 | 12.25 |

4 | 4/10 , RMB20 ; 6/10, RMB16 | 4/10 , RMB 38.5 ; 6/10, RMB 1 | 17.6 | 16 |

5 | 5/10 , RMB20 ; 5/10, RMB16 | 5/10 , RMB 38.5 ; 5/10, RMB 1 | 18 | 19.75 |

6 | 6/10 , RMB20 ; 4/10, RMB16 | 6/10 , RMB 38.5 ; 4/10, RMB 1 | 18.4 | 23.5 |

7 | 7/10 , RMB20 ; 3/10, RMB16 | 7/10 , RMB 38.5 ; 3/10, RMB 1 | 18.8 | 27.25 |

8 | 8/10 , RMB20 ; 2/10, RMB16 | 8/10 , RMB 38.5 ; 2/10, RMB 1 | 19.2 | 31 |

9 | 9/10 , RMB20 ; 1/10, RMB16 | 9/10 , RMB 38.5 ; 1/10, RMB 1 | 19.6 | 34.75 |

10 | 10/10 , RMB20 ; 0/10, RMB16 | 10/10 , RMB 38.5 ; 0/10, RMB 1 | 20 | 38.5 |

Choice in which Subject Switched to Gamble B | Proportion of Choices |
---|---|

1 | 0 |

2 | 0 |

3 | 0.07 |

4 | 0.15 |

5 | 0.25 |

6 | 0.16 |

7 | 0.26 |

8 | 0.08 |

9 | 0.04 |

10 | 0 |

**Table A3.**The Relationship between the Depth of Thinking in the P Beauty Contest Game and the 11–20 Game.

Dependent Variable: Depth of Thinking in P Beauty Contest game | |
---|---|

Depth of Thinking in the 11–20 Game | −0.02 |

(0.11) | |

Constant | 3.31 ^{***} |

(0.32) | |

R-squared | 0.001 |

Number of Observations | 92 |

## Appendix B

## Appendix C

**Instructions**[P Beauty Contest Game]

**My Decision**

**Instructions**[11–20 Game]

**My Decision**

## References

- Keynes, J.M. The General Theory of Interest, Employment and Money; Palgrave Macmillan: London, UK, 1936. [Google Scholar]
- Nagel, R. Unraveling in guessing games: An experimental study. Am. Econ. Rev.
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1 | Alaoui and Penta (2015) [7] present a model in which the player’s depth of thinking is endogenously determined. In their approach, individuals act as if they follow a cost-benefit analysis. Our approach is related to their approach in the sense that players face a trade-off over whether to forego a higher fixed payoff (cost) for the possibility of obtaining the reward (benefit). |

2 | Unfortunately, this result cannot be generalized to the general case that allows arbitrary utility functions or allows arbitrary distribution of the risk aversion level. For example, for the CARA utility, it can be verified that the equilibrium thresholds may not be monotonic. |

3 | In the literature, an alternative method for classifying the depth of thinking is to use 50 as a reference point for level 0. We do not use 50 as the reference point because doing so would require dropping data points above 50. Nevertheless, our result remains qualitatively the same, and significant, if we use 50 as the reference point. |

4 | Highly risk-averse subjects are defined as those who switch from gamble A to gamble B in choice 8 or later (i.e., the subject has chosen 7 safe choices (gamble A)). Our design very closely follows that of Holt and Laury (2002) [4]. This group of subjects is also described as very risk averse by Holt and Laury (2002) [4]. Table A1 (online appendix) reports the expected value of the gambles, assuming that the subjects take the objective probability as given. We can observe that, if a subject is risk neutral, then he should switch from gamble A to gamble B starting with choice 5. Thus, an individual who switched to gamble B at choice 8 or later must be highly risk averse. Holt and Laury (2002) [3] estimate the coefficient of relative risk aversion of their subjects using the utility function u(x) = x1 − r/(1−r) for x > 0. It is found that the coefficient of relative risk aversion increases with the number of safe choices. For example, when the subject switched at choice 8, the implied range of relative risk aversion is 0.68 < r < 0.97; thus, they classify the subject as “very risk averse”. |

5 | The risk premium of subjects who switched to gamble B in choice n is equal to [(the expected value of gamble A in choice n—the expected value of gamble B in choice n) + (the expected value of gamble A in choice (n−1)—the expected value of gamble B in choice (n−1))]/2. |

**Figure 2.**The Relative Frequencies of the Chosen Numbers and the Depth of Thinking in the P Beauty Contest Game. (

**A**) The Relative Frequencies of the Chosen Numbers in the P-beauty Contest Game; (

**B**) The Depth of Thinking in the P Beauty Contest Game.

Number | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|

Equilibrium when players are risk neutral (%) | 0 | 0 | 0 | 0 | 25 | 25 | 20 | 15 | 10 | 5 |

Equilibrium when players have utility u(x) = x^{1−r}/(1 − r) with r = 1.37 (%) | 0 | 0 | 0 | 0 | 0 | 22.2 | 30.5 | 23.3 | 16 | 8 |

Equilibrium when players have utility u(x)=1−e^{αx} with α = 0.15 (%) | 0 | 0 | 0 | 0 | 0 | 0 | 20 | 38.1 | 27.2 | 14.7 |

Experimental Results by Arad and Rubinstein (2012) (%) | 4 | 0 | 3 | 6 | 1 | 6 | 32 | 30 | 12 | 6 |

Our Experimental Results (%) | 0 | 0 | 0 | 7.3 | 1 | 4.2 | 19.8 | 37.5 | 26 | 4.2 |

Dependent Variable: Low Depth of Thinking | Dependent Variable: Depth of Thinking | |||
---|---|---|---|---|

(1) P Beauty Contest Game | (2)11–20 Game | (3)P Beauty Contest Game | (4)11–20 Game | |

Risk Premium^{5} | 0.00004 | −1.99e^{−}06 ^{***} | ||

(0.00005) | (7.27e^{−}07) | |||

High Risk Aversion | 0.04 | 0.27 ^{***} | ||

(0.13) | (0.09) | |||

Constant | 6.34 ^{***} | 2.34 ^{***} | ||

(2.10) | (0.16) | |||

R-squared | 0.04 | 0.02 | ||

Pseudo R-squared | 0.001 | 0.03 | ||

Number of Observations | 96 | 96 | 89 | 89 |

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**MDPI and ACS Style**

Li, K.K.; Rong, K.
Choices in the 11–20 Game: The Role of Risk Aversion. *Games* **2018**, *9*, 53.
https://doi.org/10.3390/g9030053

**AMA Style**

Li KK, Rong K.
Choices in the 11–20 Game: The Role of Risk Aversion. *Games*. 2018; 9(3):53.
https://doi.org/10.3390/g9030053

**Chicago/Turabian Style**

Li, King King, and Kang Rong.
2018. "Choices in the 11–20 Game: The Role of Risk Aversion" *Games* 9, no. 3: 53.
https://doi.org/10.3390/g9030053