Outcome Luck Assessment and the Luck Equation

Abstract

Luck—be it good or bad—enters when an uncertain, unforeseen outcome is beneficial (or detrimental) to someone. Its assessment is a matter of the extent to which unforeseeable developments impinge on people’s interests. The calculus of luck fuses the calculus of probability with value theory to provide a means for assessing the benefit (or detriment) that chancy evaluations provide for people. Any adequate account of the matter must determine not only whether and how it is that an agent is lucky or unlucky but must also enable us to tell to what extent this is so. It must, in sum, be quantitative. And the Calculus of Luck is the project of measuring extent to which the outcome of a chance eventuation represents good (or bad) luck for the individuals involved.

Question Appendix

• Question 1: Under what conditions are all of the possible outcomes of a chancy situations equally lucky?

  Answer: If λ{O_i} = c uniformly for all i, then uniformly |O_i| − E = c and thereby |O_i| = E + c = constant. Thus only when all outcome-yields are identical will all outcomes be equally lucky. Probabilities do not enter into it then.

• Question 2: What is the sum total of outcome luck afforded by a chancy outcome situation?

  Answer: $\sum \limits_{\mathrm{i}}^n\left(\left|{O}_{\mathrm{i}}\right|-E\right)=\sum \limits_{\mathrm{i}}\left|{O}_{\mathrm{i}}\right|-n\times E$. So when we have is a fair situation (E = 0), then the luck total will equal the yield total. Otherwise things get complicated.

• Question 3: Can it happen that (a) Every outcome of a chancy situation is lucky, or (b) that no outcome is lucky.

  Answer: (a) Clearly not. It cannot be that λ{O_i} = |O_i| − E is always > 0. For this would require that all |O_i| are > E which is impossible given E’s definition. (b) However, when all outcomes have equal yield then none will be lucky.

• Question 4: Can it happen that (a) Every outcome of a chancy situation is unlucky, or (b) that no outcome is unlucky?

  Answer: (a) clearly not. It cannot be that λ{O_i} = |O_i| − E is always < 0. For this would require that all |O_i| are < E which is impossible given E’s definition. (b) However, when all outcomes have equal yield then none will be lucky.

• Question 5: Under what condition will an outcome-yield in a chancy situation equal the expectation of that situation itself?

  Answer: Consider a situation of the following format:

From the specification of E{S} it follows that E{S} = p × E{S} + (1 – p) × Z, and so: E{S} = Z. The value of p become irrelevant, and Z = E{S}. The answer to our question in thus: as long as all the outcomes of the situation are identical.