Game Theory #2

Hello again!

In part #1 we had a very brief introduction of what is Game Theory but now is time to look into “the games” and how to represent the utility. The most basic kind of game, sort of “the standard” are called Normal-Form games, also Strategic-Form or Matrix-Form but those are more fancy names.

How’s the Normal-Form (NF) like?

• Is a game with a finite number n of players.
• It uses an n-dimensional matrix, which is the same to say a table (or “tuple”).
• Represents the utility for every player, for every state of the world.

Now, you see highlighted the “state of the world” part. Just another fancy term that we will understand at the end of the post, I promise. But first:

Utility theory: as the title implies, it’s a theoretical approach (which basically means that is based on a reasoning or system of principles to describe or explain something) that quantifies the degree of preference an agent has among a set of available alternatives. e.g.

e.g. You (agent) can have for dinner pizza, a salad, or cake. Those are a set of available alternatives and the utility theory says “hey man! how much do you like each?” Well (you reply), on a scale from 1 to 10 I like pizza 8, salad 5, and cake 10.

	Pizza	Salad	Cake
You	8	5	10

Those values (8,5,10) represent the level of happiness or utility you get from each food. If you have to decide which one you’re having for dinner, as a rational agent it’ll be pretty easy; you’ll choose the cake because is the one that makes you happier. Noticed that it’s easier to decide when you look at real numbers? Just remember that a game’s outcome is not always what the players intended.

Here’s another term we’ll use:

Utility function: states of the world mapped into real numbers.

We keep saying “state of the world” and again, it’s nothing but a term and the final key to read a game in NF.

Behold, the ancient game in the book!!! (actually, no but probably to most well known)

The Prisoner’s Dilemma

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge. They hope to get both sentenced to a year in prison on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The offer is:

• If A and B each betray the other, each of them serves 2 years in prison
• If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison (and vice versa)
• If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge

*We call a (say with me) “state of the world” to any of the possible situations listed above.

This is how it looks in NF

By Kilom691 – Own work, CC BY-SA 4.0, Link

• For our players A and B.
• The available alternatives for each player are C (cooperate) and D (defect/betray), also called action profiles.
• The utility can be either from [0,1,2,3] (in this case the lower the better).

You see, depending on the action profiles chosen by the players, the state of the world could be [C,C], [C,D], [D,C], [D,D]. Each of them with its own possible outcomes.

Give it a look and we’ll meet on part #3 to analyse the outcomes  🙂

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Prisoner’s Dilemma blatantly taken from the  Wikipedia page

Extra: in case you have spare time to read something online, here’s a pdf I found on the first page of my search on Theoretical Approach: How are theoretical approaches expressed in research practices?