coefficient of 1.9×10−3. Those estimates are the order of magnitude smaller than those estimated from lottery choice experiments. For example, Choi et al. (2007) obtain a median estimate of 0.029, Holt and Laury (2002) have 0.032, and von Gaudecker et al. (2011) obtain 0.018 from a laboratory experiment and 0.032 from an online survey.8
Taken together, the magnitude of absolute risk aversion estimated in the cur- rent study is consistent with those find in the field settings including other TV game shows and insurance choices, but they are much closer to risk-neutral com- pared to typical utility function estimated from lottery choice experiments.
and is applied to every situation except for some very special cases. Under this rule, player 1 is suggested to wager to cover a doubled score by player 2, i.e., wager y1 =2x2−x1or $1 more of that. This corresponds exactly to the “shutout bet” in Metrick’s (1995) terminology. For player 2, folk strategies depend on the threshold number m ∈ N but folk strategies can be expressed as intervals of the form[wm2, ¯wm2]. Whenm =1, it is suggested that player 2 should wager everything (i.e., y2 = x2). When m = 2, it is suggested that player 2 should wager between w22 = 0 and ¯w22 = 3x2−2x1. When m = 3, it is suggested that player 2 should wager between w32 = x1−x2 and ¯w32 = 3x2−2x1. When m ≥ 4, it is suggested that player 2 should wager between w42 =2(x1−x2)and ¯w42 =3x2−2x1.
Notice that the value of common upper bound 3x2−2x1 comes from the fact that player 1’s final score would become 2x1−2x2 if she wagered y1 = 2x2−x1
and her answer was incorrect. Player 2’s wager y2 ≤ 3x2−2x1 guarantees her final score being at least this amount even if her answer was incorrect.
The main objective of the analysis in this section is how often each player obeys those suggested folk strategies. Tables 4.9 and 4.10 present frequencies of wagers falling into each category for each value of m ∈ {1, 2, 3, 4}. First, the majority of player 1 chose to wager more than 2x2−x1 to achieve a higher score than player 2’s maximum possible score. However, many of those bets are within $100 from the threshold. Player 1 rarely bet $0, and the frequency of the bet somewhere between $0 and 2x2−x1increases asmbecomes large (i.e., the difference between player 1’s score and player2’s score shrinks).
In contrast with player 1’s general tendency to obey the folk strategy, player 2’s bets fall outside the suggested ranges quite often (Table 4.10). Except for the case of m = 1, where “bet everything” (i.e., y2 = x2) is the suggested strategy, player 2 tends to overbet.
We then ask which subset of player 2 chose to obey folk strategies and who chose to overbet and which subset of player 1 chose to bet the amount that is sufficient to cover player 2’s maximum possible score and who chose to bet more aggressively. It is reasonable to hypothesize that players base their decisions on their subjective beliefs. The general regression framework to capture this effect can be written as
Pr[ξi =1|Zi] =G(Zi,β),
where ξi is an indicator variable that takes on the value 1 if the i-th observation of player 2 chooses y2 ∈ (w¯m2,x2] and 0 if y2 ∈ [wm2, ¯w2m] for m ≥ 2 (we thus drop observations where player 2 chose to underbet) for the case of player 2, and similarly, takes on the value 1 if the i-th observation of player 1 chooses
Table4.9: Frequencies of player 1’s wager categories.
m=1 m=2 m =3 m ≥4
y1 =0 0.012 0.018 0.013 0.009
y1 ∈ (0, 2x2−x1) 0.051 0.125 0.139 0.198 y1 =2x2−x1 0.060 0.049 0.040 0.045 y1 ∈ (2x2−x1, 2x2−x1+100] 0.600 0.584 0.613 0.531 y1 >2x2−x1+100 0.278 0.225 0.195 0.216
N 1,004 570 375 1,495
Note: Italicized numbers represent the frequencies with which observed wagers match folk strategies suggestions.
Table4.10: Frequencies of player 2’s wager categories.
m=1 m=2 m =3 m≥4 y2∈ [0,wm2) – – 0.120 0.199 y2∈ [wm2, ¯w2m] 0.014 0.156 0.245 0.363 y2∈ (w¯m2,x2) 0.736 0.675 0.488 0.280 y2= x2 0.250 0.168 0.147 0.157
N 1,004 570 375 1,495
Notes: For m = 1, we define w12 = w¯12 = 0. Italicized numbers represent the frequencies with which observed wagers match folk strategies suggestions.
y1 ∈ [2x2−x1, 2x2−x1+100] and 0 ify1 ∈ (2x2−x1+100,x1]for m≥2 (similar remark applies). The vectorZicomprises a set of variables includingScore xi, Diff x1−x2, and a dummy variableMale. Since we are primarily interested in whether estimated subjective beliefs can rationalize their wagering decisions, we include Belief as well as Belief2 to capture potential nonlinear effect. We choose a logit specification for the function G.
Table 4.11 presents result from a series of logistic regressions. In specifica- tion (2), the coefficient onBelief is negative and significant, and the coefficient on Belief2 is positive and significant. Those two coefficients are close in magnitude (the absolute values are not significantly different each other, p = 0.657). Those observations imply that the relationship between probability of overbetting and subjective belief is U-shape with the reflection point around 0.5. In other words, players who are more “confident,” in both positive and negative directions, tend to wager more than what folk strategies suggest. Columns (4) to (6) indicate that the model does not fit player 1’s behavior well. The coefficients on Belief and Belief2 suggestU-shaper relationship but neither of them are significant. Further-
Table4.11: Estimation result.
Player 2 Player 1
(1) (2) (3) (4) (5) (6)
Belief −18.80 −65.52∗ −70.46∗ −17.21 −23.11 −21.54
(29.22) (31.69) (31.82) (21.75) (21.60) (22.05)
Belief2 12.28 63.72∗ 67.94∗ 16.58 23.65 22.44
(28.10) (30.54) (30.64) (21.42) (21.29) (21.70)
Belief−i 28.00 −21.79
(21.23) (27.58)
Belief2−i −25.68 17.53
(21.09) (26.77)
Score(K) −0.132∗∗∗ −0.136∗∗∗ −0.003 0.005
(0.015) (0.016) (0.014) (0.016)
Diff (K) 0.491∗∗∗ 0.479∗∗∗ −0.068 −0.059
(0.040) (0.041) (0.036) (0.045)
Male −0.030 −0.032 0.143 0.143
(0.104) (0.104) (0.110) (0.111)
Constant 7.045 17.79 11.73 5.493 6.746 12.68
(7.576) (8.198) (9.285) (5.508) (5.472) (8.619)
N 1,983 1,983 1,983 1,987 1,987 1,987
PseudoR2 0.009 0.096 0.097 0.000 0.003 0.005
Notes: For player 2, the dependent variable is a dummy that takes 1 if y2 ∈ (w¯m2,x2] and 0 if y2 ∈ [wm2, ¯wm2]. For player 1, the dependent variable is a dummy that takes 1 if y1 ∈ [2x2−x1, 2x2−x1+100] and 0 if y1 ∈ (2x2−x1+100,x1]. Only observations in m≥2 are included. Robust standard errors are in parentheses. Stars indicate significance level. ∗∗∗: p<0.001,∗∗: p<0.01,∗: p<0.05.
more, we do not observe any effect of own score nor score difference.
Those results, taken together, imply that player 2’s wagering decisions are in part based on their subjective beliefs, while player 1’s are not. It might be the case that player 1 tends to choose some amount close to 2x2−x1 mechanically, rather than deliberately think about her own chance of answering correctly.