One marvelous example of the conflict between mathematics and human behaviour is shown in the “Allais Paradox.” Compared to probability theory, in the Allais Paradox, people choose correctly or incorrectly based on irrelevant details.
Probability, Payout, Expected Value and Lotteries
The mathematical view of “probability” is the likelihood that some specific outcome will occur from an event. Some events might result in a benefit to a participant or observer. A game or lottery has some outcomes classed as “wins” or “losses”. A “win” might indeed result in some benefit, whether it is a cash prize to a participant or the settlement of a gambling bet. The “payout” is the benefit to the winner. This payout may depend on the exact outcome. For example, $1 for every point in the margin of victory in the score between two sports teams.
The “expected value” or “expected utility” of an event is the sum of the payout for each outcome multiplied by the probability of that outcome.
An example should make this clear. Let’s toss a coin twice, so the outcomes are two heads (HH), two tails (TT) or one head and one tail (HT or TH). Each outcome is equally probable; ‘p’ = 0.25. The only payout is $4 for a double head (HH). So the expected value of one event is 0.25 * $4 = $1.
By the way, this means that a fair payment to play this game is $1, the expected value of a single event.
A “lottery” is a game of chance in which the outcome is a unique item drawn from a pool of resources, and each item has the same probability of being drawn. Only one item is drawn, so it is impossible for one lottery to have two or more outcomes.
The double coin toss is equivalent to a lottery drawing the numbers 1, 2, 3, or 4, and paying $4 only when the number 1 is drawn.
Lottery’s Independence Axiom in Informal Attire
This does require a somewhat formal beginning, but then this axiom will slip into something more comfortable.
Let’s say that a lottery has at least three outcomes, called ‘X’, ‘Y,’ and ‘Z’. The observer’s preference for outcome ‘X’ over ‘Y’, is shown as “X > Y”. This means that the observer’s payout is larger for ‘X’ than for ‘Y’. Conversely, if the payouts are the same for ‘X’ and ‘Y’, then the observer is “indifferent”. That is shown as “X ~ Y”.
In a mathematical lottery, an observer can always rank preferences and indifferences. There is no “rock, paper, scissors” lottery, with a loop wherein “rock” > “scissors” > “paper” > “rock”…
In probability theory, ‘p’ is the probability of an outcome. It is always the case that 0 <= p <= 1.
In lottery theory, given “X > Z”, the construction “p*X + (1-p)*Z” means that you can get any outcome ‘Y,’ whose desirability lie between ‘X’ and ‘Z’.
The Formal Independence Axiom
For all outcomes X, Y and Z, if X > Y, then for all ‘p’ where 0 <= p <= 1, p*X + (1-p)*Z > p*Y + (1-p)*Z .
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