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## Game Theory Introduction

The Roots, 2006

A game is ‘any interaction between agents that is governed by a set of rules specifying the possible moves for each participant and a set of outcomes for each possible combination of moves’ (Hargreaves and Varoufakis, 2004 p. 3)

Aim: describe rational behaviour in

Wouldn’t it be cool if we had a way of saying, for any situation, how interacting rational agents would act? Suppose we could specify a general recipe which would tell us, for any situation at all, which actions rational agents would perform. Wouldn’t that be useful to understanding social interactions?

When two or more agents interact,
so that which outcome one agent’s choice brings about depends on how another chooses,
how do their preferences guide their choices?

first the representation

Here is a very simple situation in which you face a choice. Assuming you prefer £10 to £10, I can predict which box you will open.
Alternatively, observing which box you open will tell me whether you prefer to have £10 or £0. (significance [for later]: Revealed Preference interpretation)
 me put £10 in box A put £10 in box B you open box A £10£0 £0£0 open box B £0 £0 £10£0
But now consider this situation. Here it is not just your choice that determines the outcome, but also mine. But you don’t have an information about what I will do. So now there’s nothing for you to do but pick a box at random.
The tacit assumption is that I am getting nothing no matter what I do. But suppose we change the game slightly ... suppose you offer me £2 to put the money in box A. Then your situation changes ...
 me put £10 in box A put £10 in box B you open box A £8 £2 £0 £0 open box B -£2 £2 £10 £0
If I prefer £2 to £0[, and if I am rational, and ...] then I will put the money in box A. If you know all this, you can predict my action. And if you can predict my action, you can rationally choose to open box A.
Let me pause over this. Suppose that I don’t care about your reward, only my own. Suppose also that I prefer £2 to £0. Then regardless of what you do, I should put the money into box A
This is a relatively simple interaction: the outcome my actions bring about for me does not depend at all on what you do.
By contrast, which outcomes your actions bring about depends on what actions I select.
Note also that it is rational for you to choose box A even if your preferred outcome would be to get £10 rather than £8. As a rational agent, you want to best satisfy your preferences. But of course you can’t just follow the money: instead you have to take into account how I am likely to act.

How you act
is a function of two things:

 me put £10 in box A put £10 in box B you open box A £8 £2 £0 £0 open box B -£2 £2 £10 £0
I just argued that it is rational for you to open box A. What assumptions are needed to establish that it is rational for you to choose to open box A? For example, you have to assume that I prefer £2 to £0. [We’ll come back to these later, after explain the programme behind Game Theory]

Game theory builds on decision theory

preferences

Your preferences impose a ranking on outcomes. (Ignore cardinal preference functions for now.)

To exhibit instrumental rationality is to select those actions which you expect to best satisfy your preferences.

To exhibit \emph{instrumental rationality} is to select those actions which you expect to best satisfy your preferences. (See Osborne & Rubinstein (1994, p. 4) for a concise and simple formal presentation.)
Don’t want to make assumptions about preferences for money, nor that money is involved in all games.
You also shouldn’t assume that I am indifferent to whether you gain £8. Maybe I want to punish you more than I want to gain £2; or maybe I am really nice and I want you have the money.
 me put £10 in box A put £10 in box B you open box A £8 £2 £0 £0 open box B -£2 £2 £10 £0
We therefore normally represent outcomes in terms of preferences (or payoffs) rather than money.

Why study game theory and its limits?

Consider three perspectives ...
[Used to say four perspectives, did I miss one?]
\subsection{Why game theory?}

‘game theory is the most important and useful tool in the analyst's kit whenever she confronts situations in which

what counts as one agent's best action (for her)

depends on

expectations about what one or more other agents will do,

and

what counts as their best actions (for them) similarly

depend on

(Ross, 2018).

Actually, in the game we considered so far this is not true. What your best action is does depend on what you expect me to do. But this is not true for me.
 me put £10 in box A put £10 in box B you open box A £8 £2 £0 £0 open box B -£2 £2 £10 £0

terminology: dominance

strict dominance dominance weak dominance
ok, so we need a different game to illustrate this.

Why study game theory and its limits?

\subsection{Why game theory?}

‘game theory is the most important and useful tool in the analyst's kit whenever she confronts situations in which

what counts as one agent's best action (for her)

depends on

expectations about what one or more other agents will do,

and

what counts as their best actions (for them) similarly

depend on

(Ross, 2018).

Here’s a game where our actions do have that kind of interdependence.
[previously used game_far_near but that is a terrible illustration]
Hawk-Dove

 Gangster X stay home attack Gangster Y stay home 22 13 attack 31 00
Why not simply decision theory? Because we cannot assign subjective probabilities

Why not simply decision theory?

How you act
is a function of two things:

are a function of your knowledge of two things:

Consider all this complexity. You cannot model this with decision theory alone.
The amazing insight behind game theory is that you can, in effect, completely remove subjective probability from the picture. Only preferences are needed.
Decision Theory is about how individuals decide which of several available actions to perform (Jeffrey, 1983, p. textbook:). Game Theory is a development which focusses on how interacting individuals select actions when which outcome one individuals’s action brings about depends on how another acts.

decision theory

How do rational agents decide which of several available actions to perform?

game theory

When two or more agents interact,
so that which outcome one agent’s choice brings about depends on how another chooses,
how do their preferences guide their choices?

Why not simply decision theory? Because we cannot assign subjective probabilities

Why not simply decision theory?

How you act
is a function of two things:

are a function of your knowledge of two things:

Consider all this complexity. You cannot model this with decision theory alone.
The amazing insight behind game theory is that you can, in effect, completely remove subjective probability from the picture. Only preferences are needed.

Why study game theory and its limits? (ctd)

Osborne and Rubinstein offer a 3rd person perspective: understanding others’ behaviour.

‘we treat game theory not as a branch of mathematics but as a social science whose aim is to understand the behavior of interacting decision-makers

(Osborne & Rubinstein, 1994, p. 2; compare Dixit, Skeath, & Reiley, 2014, pp. 36--7)

Compare (Dixit et al., 2014, pp. 36--7) who propose that game theory allows us to explain why people act as they do in social interactions, to predict their actions, and to find normatively correct courses of action to take.

‘understanding why game theory does not, in the end, constitute the science of society (even though it comes close) is terribly important in understanding the nature and complexity of social processes’

(Hargreaves-Heap & Varoufakis, 2004, p. 3)

short essay question:

What is team reasoning?

Which, if any, social interactions are better modeled by team reasoning than game theory?

plan

1. What is game theory?

2. What are its limits?

3. What is team reasoning and how might it overcome the limits?

Here we still are because we have only explained what games are and how they are represented.
No theory yet.