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Applications and Limits of Game Theory

theory is supposed to explain things. Which things?

‘Many events and outcomes prompt us to ask: Why did that happen? [...] For example, cutthroat competition in business is the result of the rivals being trapped in a prisoners’ dilemma’

(Dixit, Skeath, & Reiley, 2014, p. 36).

(Dixit et al., 2014, p. 36)

Actually just this one game which we looked at earlier has a striking range of applications ...
Prisoner X
resistconfess
Prisoner 59640resist3
3		0
4
confess 4
0		 1
1

‘Many events and outcomes prompt us to ask: Why did that happen? [...] For example, cutthroat competition in business is the result of the rivals being trapped in a prisoners’ dilemma’

(Dixit et al., 2014, p. 36).

(Dixit et al., 2014, p. 36)

pd is a good target because common.

Games with the Prisoner’s Dilemma structure arise in:

bower birds (maraud/guard nests)

business organisations (product pricing)

countries (international environmental policy)

individual adult humans (suspects under arrest)

(Dixit et al., 2014, p. chapter 10)

Why is this true?
It’s (a) the game they are playing has the structure of a prisoner’s dilemma; and (b) each rationally avoids the strictly dominated actions, resulting in a worse outcome overall for both than would be possible if they could collude.

?

game theory

Aim: describe rational behaviour in .

How you should act (in a noncooperative, one-off game):

  • avoid dominated actions
  • select any Nash equilibrium
Nash equilibrium allows us to identify rationally optimal actions in a way that doesn’t involve working through how these beliefs might be formed.

Entails:

Resisting (‘cooperating’) is not rational in the Prisoner’s Dilemma.

Choosing ‘Low’ in Hi-Low is rational.

game theory explains why things happen

Here we’re tacitly considering a game played repeatedly, and we’re edging into evolutionary game theory.

further illustration : side-botched lizzards

(Sinervo & Lively, 1996)

Key to the explanation is that the three types of male have different strategies: ultradominant—large territory; pacific—small teritory; sneaker—no territory.
If one type becomes scarcer in one year, individuals will be super successful in mating and so produce more offspring the next.

(Sinervo & Lively, 1996)

The important thing from game theory is that rock-paper-scissors has no Nash equilibrium. That’s why all three strategies persist. (Compare playing rock-paper-scissors against someone who always picked the same thing: would be easy to win, no?)
Player X
rock paperscissors
Player Yrock0
0		-1
1		1
-1
paper 1
-1		 0
0		-1
1
scissors -1
1		 1
-1		0
0
‘Each of two people chooses either Head or Tail. If the choices differ, person 1 pays person 2 a dollar; if they are the same, person 2 pays person 1 a dollar. Each person cares only about the amount of money that he receives. A game that models this situation is shown in Figure 17.3. Such a game, in which the interests of the players are diametrically opposed, is called “strictly competitive”. The game Matching Pennies has no Nash equilibrium’ (Osborne & Rubinstein, 1994).
Can you identify the nash equilibira?
There are none!

Sinervo & Lively (1996)

I couldn’t resist this one ... game theory (rock-paper-scissors specifically) has been used to explain ‘evolutionary stable strategy model to a three-morph mating system in the side-blotched lizard’ (Sinervo & Lively, 1996). (The ones on the right resemble sexually receptive females morphologically; they are ‘sneakers’.)

aside: Where do the preferences come from?

limited range of actions

∴ not decision theory

In this case we are not relying on decision theory (limited range of actions)

task : find cases where game theory explains things

  • in law: inequality, culture and power (McAdams, 2008)
  • network security (Roy et al., 2010)
  • evolution of social contract (Skyrms, 2000)
    • ‘evolutionary models supply a rationale for Nash equilibrium that rational choice theory is hard-pressed to deliver. Furthermore, in cases with multiple symmetrical Nash equilibria, the dynamic models offer a plausible, historically path-dependent model of equilibrium selection’ (Skyrms, 2000)
  • distribution of water resources (Madani, 2010)
  • the tragedy of the commons (Tadelis, 2013, p. §5.2.2)
  • foraging behaviours (Hansen, 1986)
  • ...
Interesting question about each of these cases: What is it that has preferences? On what basis are the preferences assigned?

how to describe a case

1. specify the game
[rock, paper, scissors]

2. map the observed behaviours onto the game
[aggressive beats subordinate, subordinate beats ...]

3. specify the principle needed
[because this game has no nash equilibrium, no action can win]

Just here we are moving from applications to limits (with hardly a break ...)

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? ✓

1a. What are its applications? ✓

2. What are its limits? ✓

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

?

game theory

Aim: describe rational behaviour in .

How you should act (in a noncooperative, one-off game):

  • avoid dominated actions
  • select any Nash equilibrium
Nash equilibrium allows us to identify rationally optimal actions in a way that doesn’t involve working through how these beliefs might be formed.

Entails:

Resisting (‘cooperating’) is not rational in the Prisoner’s Dilemma.

Choosing ‘Low’ in Hi-Low is rational.

Or does it?
weak dominance strict dominance dominance
Start by explaining dominance (simpler than a nash equilibrimm)
Prisoner X
resistconfess
Prisoner 59640resist3
3		0
4
confess 4
0		 1
1
observation: people in this kind of situation will rationally end up performing actions which are mutually harmful in the sense that there is a better course of actions available to them.
game theory -> always get a worst outcome in pd

?

game theory

Aim: describe rational behaviour in .

How you should act (in a noncooperative, one-off game):

  • avoid dominated actions
  • select any Nash equilibrium
Nash equilibrium allows us to identify rationally optimal actions in a way that doesn’t involve working through how these beliefs might be formed.

Entails:

Resisting (‘cooperating’) is not rational in the Prisoner’s Dilemma.

Choosing ‘Low’ in Hi-Low is rational.

Player X
high low
Player Yhigh2
2		0
0
low 0
0		 1
1
This game has two nash equilibria ...

A nash equilibrium for a game is a set of actions from which no agent can unilaterally profitably deviate

(Osborne & Rubinstein, 1994, p. 14).

What should X do?

If X expects Y to choose high, X should choose high

If X expects Y to choose low, X should choose low

But what Y should do depends on what Y expects X to choose.

So what X should do depends on what Y expects X to choose.

‘We have entered an infinite regress: what it is rational for a player in a situation like X's to do depends on what it is rational for a player in a situation like X's to do.’

(Sugden, 2000, p. 181)

minor tweaks, like invoking salience, do not work.

(Bacharach, 2006, p. Chapter 1.2--1.7)

What exactly is the limit

‘it seems obvious that ‘high’ is the rational choice [...]. Apparently, something is missing from the standard theory of rational choice. But what?’

(Sugden, 2000, p. 182)

An action is rational
in a noncooperative game
if it is a member of a nash equilibrium?

Maybe this idea isn’t very plausible. It seems to get PD wrong (it says that an apparently rational course of action (resist) is nonrational) and it seems to get hi-low wrong (it says that an apparently nonrational action (low) is no less rational than an apparently rational action (high).)

?

game theory

Aim: describe rational behaviour in .

How you should act (in a noncooperative, one-off game):

  • avoid dominated actions
  • select any Nash equilibrium
Nash equilibrium allows us to identify rationally optimal actions in a way that doesn’t involve working through how these beliefs might be formed.

Entails:

Resisting (‘cooperating’) is not rational in the Prisoner’s Dilemma.

Choosing ‘Low’ in Hi-Low is rational.

how to identify a limit

1. state the fact to be explained

[people find ‘high’ the obvious choice in hi-low]

2. show that game theory cannot explain it

[simple: nash equilibria; deeper: regress]

3. consider objections

[salience as a potential fix]

‘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)

some applications of game theory succeed, others might fail

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? ✓

1a. What are its applications? ✓

2. What are its limits? ✓

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

PS

Maybe there are other kinds of eqilibrium?

(Hargreaves-Heap & Varoufakis, 2004, p. 3)
also: evolutionary game theory Skyrms (2000)