Before we move beyond zero-sum games, it's worth taking a deeper look
at the idea of utilities. As I mentioned before, in a game, the scores in
the matrix are given by something called a utility function.
Utility is an idea for how to mathematically describe preferences in terms
of a game or lottery. For a game to be valid (that is, for a game to have a meaningful analysis and solution), there must be a valid utility function that
describes the players' preferences.
But what do we have to do to make a valid utility function? It's
simple, but as usual, we'll make it all formal and explicit.
The easiest way to talk about utility and games is to think of games
as a lottery. A lottery is a game with one player, with a variable payoff. You can think of it as a game where you only have one strategy available, and
you know a probability distribution for the other player's strategy. So you
can compute a value for the game.
- Universality: Every possible outcome has a utility value
associated with it; this includes both specific final outcomes,
and indirect/meta outcomes (that is, situations where
an outcome from one lottery/game is a ticket for another lottery/game.
That outcome will have as its value the expected utility
for the lottery).
- Comparability: utility values are always comparable;
for a particular player with utility function u,
given any two choices A and B, they'll either prefer one over the
other (u(A) > u(B) or u(B)>u(A)), or they'll be indifferent (u(A) = A(B)).
There are no possible utility values that can't be compared.
- Consistency: utility comparisons are consistent. Consistency
requires that utility values be reflexive (if u(A)>u(B), then u(B)<u(A), and if u(A)=u(B) then u(B)=u(A)), and transitive (if u(A)<u(B) and u(B)<U(C) then u(A)<u(C)).
- Rationality: Players are rational actors. This means that
players will always make choices that maximize their expected payoff. So
given a lottery with expected payoff 5, and a winning of 4, a player will
always choose to play. On the other hand, people won't play when the
expected payoff is small - there's no intrinsic attraction to playing
beyond the expected payoff encoded in the utility function. (So, for example,
for a gambler who enjoys gambling, his pleasure at playing the game
is encoded in the utility value of every outcome that involves playing,
win or lose.) Even given a probabilistic situation,
where the two choices are a definite payoff of 4, or a lottery with
an expected payoff of 4, the player will be indifferent.
Let's take an example to show how we can build a consistent utility
function. Suppose we've got the following
sitaution. We've got three lotteries - A, B, and C, where:
- In lottery A, the prizes are an apple (with probability 0.3),
a pear (probability 0.2), a ticket for lottery B (probability 0.2)
- In lottery B, the prizes are a pear (probability 0.1), a banana
(probability 0.5), or a ticket for lottery C (probability 0.4).
- In lottery C, the prizes are an apple (probability 0.4), a pear
(probability 0.3), or nothing (probability 0.3).
The player can choose either a pear or a ticket for lottery A. How can
we decide what they should do?
We'll start by describing what the player prefers in the concrete prizes. They
surely don't like nothing as a prize - so we'll give that utility 0. And they like apples twice as much as pears, and bananas three times as much as pears. So
we'll assign u(pear)=1, u(apple)=2, u(banana)=3.
So what's a ticket for lottery A worth?
Since one prize of A is a ticket for B, and one prize for B is a ticket for C,
we need to figure out the utility values of tickets for B and C. We'll do C first, since it doesn't depend on the other two. The utility value of a lottery is the weighted average of its outcome utilities. So, u(c)=0.4*2 + 0.3*1 + 0.3*0 =
1.1. So a ticket for lottery C is worth just a little bit more than
a pear - given a choice between a ticket for C and a pear, the player will choose the ticket; given a choice between a ticket for C and an apple, the player
will choose the apple.
Now, we can figure out the utility value of a ticket for lottery B. u(B) =
0.1*1 + 0.5*3 + 0.4*1.1 = 2.04. So a ticket for lottery B is worth a lot more than a ticket for lottery C, and it's also worth more than either an apple or a pear.
Now, finally, we can compute the value of a ticket for A. u(A) = 0.3*2 + 0.2*1 + 0.2*2.04 + 0.3*0 = 0.6+0.2+0.408=1.208. So according to the utility function, the player will choose a ticket for lottery A.
There's a couple of properties of utility functions that can seem strange
at first. Given a utility function u, you can define another utility
function such that ∀x:v(x)=u(x)+c (where C is a constant). In every
possible game, the two utility functions are effectively equivalent: they'll result in exactly the same strategic choices. Depending on how you look at it, this can be either entirely obvious, or it can seem very strange. If you look at
the example above, where we defined initial utility values in terms of the magnitude of preference - that is, "I like an apple twice as much as a pear, so I'll say that an apple equals 2 pears", it seems strange. But the real driving
thing in the utility function is the simple difference between utilities - and the simple difference is exactly the same - so the strategies will be the same.
Similarly, utility functions have equivalent outcomes when multiplied by a constant. Again, it's the difference between things that are important, and if the distance between outcomes is varied consistently - either by adding or multiplying, then the strategic outcomes will be the same.
It's all very simple and very rational. Applying it is a lot harder - because
people aren't always rational. People will gamble for the fun of gambling
depending on their mood; an apple may be worth more than a pear today, and
less tomorrow; a person may prefer an apple rather than a pear, and be indifferent to an apple versus a banana, but prefer a pear to a banana. People just
don't necessarily define the values of their choices in ways that fit
with the utility function requirements of consistency, comparability, and
As we'll see in later posts, you can model some interesting economic phenomena using game theory and utility functions. But those models are tricky to apply in the real world, because real-world decision making is often too complex to be able to accurately model with a comprehensible, analyzable utility function. This leads to some really intense debates between political perspectives, ranging from
people who find the whole idea of describing human interactions in terms of utility functions to be morally obscene; to people who believe that there's no moral problem but that human behavior is too complex to describe as utility function; to people who believe that utility functions are the ideal way (both mathematically and ethically) to model behavior. Interestingly, the lines between these viewpoints don't correspond to the traditional political divide. Most people seem to expect that the "morally obscene" group should correspond to
the political left, and that the "utility functions are perfect" should correspond to the right. But you can find libertarians and free-market uber-conservatives on the morally obscene side, and marxists on the "utility functions are ideal" side. It's pretty interesting.