Before we move beyond zero-sum games, it’s worth taking a deeper look
\nat the idea of utilities. As I mentioned before, in a game, the scores in
\nthe matrix are given by something called a utility function.<\/p>\n
Utility is an idea for how to mathematically describe preferences in terms
\nof 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
\ndescribes the players’ preferences. <\/p>\n
But what do we have to do to make a valid utility function? It’s
\nsimple, but as usual, we’ll make it all formal and explicit.<\/p>\n
<\/p>\n
The easiest way to talk about utility and games is to think of games
\nas 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
\nyou know a probability distribution for the other player’s strategy. So you
\ncan compute a value for the game.<\/p>\n
Let’s take an example to show how we can build a consistent utility
\nfunction. Suppose we’ve got the following
\nsitaution. We’ve got three lotteries – A, B, and C, where:<\/p>\n
The player can choose either a pear or a ticket for lottery A. How can
\nwe decide what they should do?<\/p>\n
We’ll start by describing what the player prefers in the concrete prizes. They
\nsurely 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
\nwe’ll assign u(pear)=1, u(apple)=2, u(banana)=3.<\/p>\n
So what’s a ticket for lottery A worth? <\/p>\n
Since one prize of A is a ticket for B, and one prize for B is a ticket for C,
\nwe 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 =
\n1.1. So a ticket for lottery C is worth just a little bit more than
\na 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
\nwill choose the apple.<\/p>\n
Now, we can figure out the utility value of a ticket for lottery B. u(B) =
\n0.1*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.<\/p>\n
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.<\/p>\n
There’s a couple of properties of utility functions that can seem strange
\nat first. Given a utility function u, you can define another utility
\nfunction such that ∀x:v(x)=u(x)+c (where C is a constant). In every
\npossible 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
\nthe 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
\nthing 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. <\/p>\n
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.<\/p>\n
It’s all very simple and very rational. Applying it is a lot harder – because
\npeople aren’t always rational. People will gamble for the fun of gambling
\ndepending on their mood; an apple may be worth more than a pear today, and
\nless 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
\ndon’t necessarily define the values of their choices in ways that fit
\nwith the utility function requirements of consistency, comparability, and
\ntransitivity.<\/p>\n
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 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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[88],"tags":[],"class_list":["post-660","post","type-post","status-publish","format-standard","hentry","category-game-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-aE","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/660","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=660"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/660\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=660"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=660"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\npeople 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<\/em> correspond to the traditional political divide. Most people seem to expect that the “morally obscene” group should correspond to
\nthe 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.<\/p>\n","protected":false},"excerpt":{"rendered":"