MODEL OF THE ST. PETERSBURG PARADOX
We now return to the original problem. Players who would be influenced by the high payoff in the event of 31 or more tails would also visit the Leningrad casino. In the previous section we practically ruled out the possibility of rational agents visiting the Leningrad casino. We are therefore left only with those attracted by "normal" payoffs in other [less unrealistic) cases. The expected value of such a payoff is
The subjective probability of survival of an agent with income equal to k-times the subsistence level b after paying Y to enter the St Petersburg casino is not influenced by the highly unlikely, astronomically high payoffs for 31 or more tails, since the 30th tail—with a win of 230—has already guaranteed his survival with a probability of at least 99.999%.
If the coin came down heads every time in the first z tosses, the agent has income equal tod-Y+2' = k-b-Y+2'. His probability of survival in this case is
The player's expected probability of survival if the coin comes down heads no more than thirty times in a row is therefore:
The maximum acceptable fee is the one for which this expression equals
25 We could also theoretically consider an agent with income d = b + 5, where d is so small that the survival probability 8/b is smaller than the payout probability 2-31 . This case, however, can be ruled out because even for the smallest indivisible money unit (let's say one cent) it holds that 0.01/b >2-31
i.e. the probability of survival if the agent does not enter the casino:
(*)
Let us denote the left-hand side of this equation as:
As can be seen from the following Figure 5 and Figure 6, equation (*) has a single solution in the "sensible” (i.e. economically interpretable) part of the domain:
Figure 5: Plots of fk,100(Y) for k Î {1.1; 2; 5}. For k ≥ 10 the graph is virtually coincident with the x-axis on the scale used
Figure 6: Plots of f2,b{Y} for b Î {100; 200; 500}
Table 1 shows solutions of equation (*) for various values of k and b:
Table 1: Maximum acceptable St. Petersburg casino entry fee Y versus agent's economic situation (i.e. versus k) and versus extinction zone boundary b
k |
b= 100 |
b = 200 |
b = 500 |
1.1 |
6.6 |
7.45 |
8.7 |
2 |
7.35 |
8.25 |
9.5 |
3 |
7.9 |
8.8 |
10.1 |
4 |
8.25 |
9.2 |
10.5 |
5 |
8.55 |
9.5 |
10.8 |
10 |
9.5 |
10.5 |
11.8 |
20 |
10.5 |
11.45 |
12.8 |
30 |
11.05 |
12.05 |
13.35 |
40 |
11.45 |
12.45 |
13.75 |
50 |
11.8 |
12.8 |
14.1 |
60 |
12.05 |
13.05 |
14.35 |
75 |
12.35 |
13.35 |
14.65 |
100 |
12.8 |
13.75 |
14.95 |
Source: Authors' calculations
Figure 7 presents the maximum acceptable fee Y as a function of the agent's economic situation (parameter k, representing the ratio of income to the subsistence level) for three different extinction zone boundaries: b Î {100; 200; 500}.
Figure 7: The maximum acceptable St. Petersburg casino nominal entry fee Y versus the agent's economic situation k (b is the extinction zone boundary)
In all cases the function is growing and concave. The agent reacts positively (with higher demand) to improvements in his economic situation, but the rate of growth of the maximum acceptable fee slows with increasing k.
The relation between the maximum acceptable fee and the magnitude of b is also expressed by a growing concave function. This relation may seem surprising at first glance: growth in b means ceteris paribus a deterioration in the agent's economic situation, since it increases his probability of economic extinction. However, it is important to realize that the level of the threat is given by the magnitude of the relative marginand that moving to higher b while keeping k constant means not only an increase in the subsistence level, but also, (i) an equal percentage rise in income, and (ii) an "inflationary” fall in the real value of both the payoff and the entry fee. If b = 100 and d = 2b = 200, a basic (i.e. minimum) payoff of 2 represents 1% of income and an entry fee of 10 is 5% of the agent's income, whereas if b = 500 and d = 2b = 1000, the basic payoff is 0.2% of income and the same nominal entry fee 10 is 1% of the agent's income. A greater percentage of income means a greater threat, so this "inflation” leads logically to the agent accepting a higher nominal, but lower real, entry fee when faced with an greater threat (due to growth in b). This relation is shown in Figure 8, where Y(b) is the maximum acceptable nominal entry fee and (Y/d)(b) is the maximum acceptable real fee, in both cases as a function of the subsistence level b for an agent whose income is equal to double the subsistence level (k = 2).
Figure 8: The relation between St. Petersburg casino entry demand and the extinction zone boundary level for an agent with economic situation k = 2
Table 2 shows how our extinction-threat-minimizing agent reacts if the game is "cut short" by the setting of a maximum number of tosses (we denote this number by N). The calculation results show that agents with the lowest income (k < 1.1) take no account of extreme profit opportunities with a probability of 2-12, i.e. 0.02 %, whereas agents with extremely high income (k ≥ 50) react more sensitively to phenomena with a probability of 2-20, i.e. 0.0001%. This is in line with reality—poor people tend to play cheap games with relatively low payoffs, whereas wealthy people are attracted to games with high entry fees and high payoffs and are deterred by games with upper limits (see the lower right-hand area of Table 2, where the equation does not have an economically interpretable solution).
Table 2: Maximum acceptable entry fees for agents with various economic situations for a maximum of N consecutive favourable outcomes (ft = 100)
k |
N = 30 |
N= 25 |
N = 20 |
N= 15 |
N= 12 |
o II |
1.1 |
6.60 |
6.60 |
6.60 |
6.60 |
6.55 |
6.45 |
2 |
7.35 |
7.35 |
7.35 |
7.35 |
7.25 |
6.95 |
5 |
8.55 |
8.55 |
8.55 |
8.50 |
7.95 |
6.20 |
10 |
9.50 |
9.50 |
9.50 |
9.20 |
7.10 |
- |
50 |
11.80 |
11.80 |
11.55 |
- |
- |
- |
100 |
12.80 |
12.80 |
11.80 |
- |
- |
- |
Source: Authors' calculations
A survival-probability-maximizing agent therefore realistically accepts an entry fee to the St. Petersburg casino (with no limit on the number of tosses) of between $6.60 and $12.80 depending on his economic situation. It turns out to be irrelevant whether the agent takes into consideration the possibility of winning only when the coin lands heads 25 times or less, or even when the number of consecutive tails is greater than 25, not only because astronomically high payoffs have an extremely low probability, but also because for a lucky agent who wins $225 any subsequent win will not affect his (almost unity) probability of survival. For poor agents the same holds for payoffs associated with an initial series of 15 or more successful tosses of the coin.
A normal agent with an income of double the subsistence level is willing to pay around 3.7% of his income for a chance at winning. By contrast, an agent in an excellent economic situation of ten times the subsistence level is willing to pay only around 1.0% of his income.
Things are different for down-and-out agents, whose probability of survival is zero if they do not enter the casino, i.e. for agents with an incometo-subsistence-level ratio of k ≤ 1. As in the Leningrad casino, the optimal strategy of these agents is unique. Even for agents who can only afford the entry fee and nothing else (i.e. d - Y = 0], entry to the casino offers a non-zero probability of economic survival [for b = 100 the probability is 2-7, i.e. just under one per cent). Such agents have a situational attraction (negative aversion) to risk — their situation forces them to take risks (even if they are risk averse in normal situations).^{[1]}
And how does interest in entering the casino change ceteris paribus with nominal payoff amount? Suppose that the payoff is v • 2i where i is the number of successful tosses (the coin lands heads / times in a row), hence 2v is the lowest payout (when the coin lands heads on the first toss and tails on the second). We solve the equation:
We again summarize the solution of this equation for various values of v (the basic payoff amount) in a table:
Table 3: Effect of multiplying payoffs on the maximum acceptable entry fee (for k = 2 or k = 5 and b = 100)
V |
k = 2 |
k = 5 |
1 |
7.35 |
8.55 |
1.1 |
7.97 |
9.31 |
1.2 |
8.56 |
10.01 |
1.5 |
10.27 |
12.07 |
2 |
12.97 |
15.33 |
3 |
17.98 |
21.40 |
5 |
26.98 |
32.44 |
7 |
35.16 |
42.54 |
10 |
46.43 |
56.54 |
15 |
63.54 |
77.89 |
20 |
79.25 |
97.55 |
30 |
108.15 |
133.70 |
Source: Authors' calculations
In the following chart we show that both cases involve a growing and concave relation:
Figure 9: The relation between the maximum acceptable entry fee and the payoff
The operator of the St. Petersburg casino can therefore stimulate the interest of a survival-probability-maximizing agent by increasing the payoffs, but the marginal effect of such increases declines—increasing the payoffs thirty-fold will increase the acceptable entry fee only around fifteen-fold.
- [1] Such agents are like the penniless man in the anecdote who, after enjoying a lavish feast in a fancy restaurant, asks whether he might pay in pearls. The head waiter says yes, so the man tells him, “Okay, bring me two more oysters and keep your fingers crossed”. In other game situations, situational risk attraction can of course arise in less absurd cases. See Hlaváček, J. et al.: Mikroekonomie sounáležitosti se společenstvím. Praha: Karolinum, 1999, 106–8.