How To Draw Normal Form Game With 3 Players

Epitomize

In the previous chapter we discussed:

Interactive decision making;
All-encompassing form games;
Extensive form games and representing information sets.

We did this looking at a game chosen "the battle of the sexes":

Celine and Bob with Information Set.

Can we think of a better way of representing this game?

Normal class games

Another representation for a game is called the normal form.

Definition of a normal form game

A $N$ role player normal form game consists of:

A finite gear up of $N$ players;
Strategy spaces for the players: $S_1, S_2, S_3, \dots S_N$;
Payoff functions for the players: $u_i:S_{i}\times S_2\dots\times S_N\to \mathbb{R}$

The convention used in this class (unless otherwise stated) is that all players aim to choose from their strategies in such a mode equally to maximise their utilities.

A natural way of representing a two histrion normal class game is using a bi-matrix. If we presume that $N=2$ and $S_1=\{r_i\;|\;1\leq i\leq m \}$ and $S_2=\{c_j\;|\;one\leq j\leq north \}$ and then a bi-matrix representation of the game considered is shown.

$A bi matrix for $N=2$. \label{L02-img01}$

Some examples

The battle of the sexes

This is the game we've been looking at between Bob and Celine:

Prisoners' Dilemma

Presume ii thieves have been defenseless by the police and separated for questioning. If both thieves cooperate and don't divulge any data they will each get a curt sentence. If one defects he/she is offered a deal while the other thief will get a long sentence. If they both defect they both get a medium length sentence.

Hawk-Dove

Suppose 2 birds of prey must share a limited resource. The birds can human activity like a hawk or a dove. Hawks always fight over the resource to the bespeak of exterminating a fellow hawk and/or take a majority of the resource from a dove. Two doves can share the resource.

Pigs

Consider ii pigs. One ascendant pig and 1 subservient pig. These pigs share a pen. There is a lever in the pen that delivers food but if either pig pushes the lever information technology will take them a little while to get to the food. If the dominant pig pushes the lever, the subservient sus scrofa has some fourth dimension to swallow most of the food before existence pushed out of the way. If the subservient pig button the lever, the ascendant pig volition eat all the food. Finally if both pigs go to push the lever the subservient pig will be able to eat a third of the food.

Matching pennies

Consider two players who can choose to display a coin either Heads facing up or Tails facing up. If both players show the same face then player one wins, if not then player two wins.

Mixed Strategies

And so far we have only considered so called pure strategies. We will now allow players to play mixed strategies.

Definition of a mixed strategy

In an $N$ player normal course game a mixed strategy for player $i$ denoted by $\sigma_i\in[0,1]^{|S_i|}_{\mathbb{R}}$ is a probability distribution over the pure strategies of player $i$. Then:

For a given player $i$ we denote the set of mixed strategies equally $\Delta S_i$.

For instance in the matching pennies game discussed previously. A strategy profile of $\sigma_1=(.2,.viii)$ and $\sigma_2=(.six,.4)$ implies that actor one plays heads with probability .2 and player two plays heads with probability .6.

Nosotros can extend the utility office which maps from the gear up of pure strategies to $\mathbb{R}$ using expected payoffs. For a two player game we accept:

(where we relax our annotation to let $\sigma_i:S_i\to[0,1]_{\mathbb{R}}$ so that $\sigma_i(s_i)$ denotes the probability of playing $s_i\in S_i$.)

Matching pennies revisited.

In the previously discussed strategy contour of $\sigma_1=(.2,.8)$ and $\sigma_2=(.6,.4)$ the expected utilities can exist calculated as follows:

Example

If we assume that player 2 ever plays tails, what is the expected utility to player ane?

Let $\sigma_1=(x,one-x)$ and nosotros have $\sigma_2=(0,1)$ which gives:

Similarly if player one always plays tails the expected utility to role player 2 is:

A plot of this is shown.

$Plot of utility if both players play tails. \label{L02-plot01}$

Add to this plot by assuming that the players independently both play heads.