Questions

Conclusions

Holt and Laury Gambles

Here are the 10 gambles from Holt & Laury (2002)

index a.x a.px a.y a.py b.x b.px b.y b.py a.ev b.ev a.var b.var ev.diff high.ev.option
1 2 0.1 1.6 0.9 3.85 0.1 0.1 0.9 1.64 0.48 0.01 1.27 1.17 a
2 2 0.2 1.6 0.8 3.85 0.2 0.1 0.8 1.68 0.85 0.03 2.25 0.83 a
3 2 0.3 1.6 0.7 3.85 0.3 0.1 0.7 1.72 1.23 0.03 2.95 0.49 a
4 2 0.4 1.6 0.6 3.85 0.4 0.1 0.6 1.76 1.60 0.04 3.38 0.16 a
5 2 0.5 1.6 0.5 3.85 0.5 0.1 0.5 1.80 1.98 0.04 3.52 -0.18 b
6 2 0.6 1.6 0.4 3.85 0.6 0.1 0.4 1.84 2.35 0.04 3.38 -0.51 b
7 2 0.7 1.6 0.3 3.85 0.7 0.1 0.3 1.88 2.73 0.03 2.95 -0.84 b
8 2 0.8 1.6 0.2 3.85 0.8 0.1 0.2 1.92 3.10 0.03 2.25 -1.18 b
9 2 0.9 1.6 0.1 3.85 0.9 0.1 0.1 1.96 3.48 0.01 1.27 -1.52 b
10 2 1.0 1.6 0.0 3.85 1.0 0.1 0.0 2.00 3.85 0.00 0.00 -1.85 b

Here are the gambles represented in a coordinate space:

Introducing competition

For each gamble pair, I calculated the probability that gamble A would beat gamble B given a one shot game. For example, for pair 4, the probability that gamble A (the safe gamble) wins is \(.40 \times 0.6 + 0.60 \times 0.60 = 0.60\). Here are the results

index ev.diff high.ev.option p.a.wins
1 1.17 a 0.9
2 0.83 a 0.8
3 0.49 a 0.7
4 0.16 a 0.6
5 -0.18 b 0.5
6 -0.51 b 0.4
7 -0.84 b 0.3
8 -1.18 b 0.2
9 -1.52 b 0.1
10 -1.85 b 0.0
  • Interestingly, it looks like the high EV gamble is always the gamble with the higher probability of winning. For pairs 1 through 4, the safe gamble is both the high EV gamble and the gamble with the highest probability of winning. For pairs 5 through 10, the risky gamble is both the high EV gamble and the gamble with the highest probability of winning.