Game Theory and Climate Change - Warwick Insitehomepages.warwick.ac.uk/~masbm/ClimateCourse/Lectures/week7:… · Game Theory and Climate Change I Climate modelling involves mathematical

Game Theory and Climate Change

David MondMathematics InstituteUniversity of Warwick

November 15th, 2013


I Climate modelling involves mathematical challenges ofunprecedented complexity. Today’s climate models

I gather data at ' 109 different locations,

I input it into a complicated partial differential equation,

I process the results

I compare with subsequent data,

I and try to improve the equationsI to make more accurate long-term predictions.

They use the world’s most powerful computers, and havebecome steadily more convincing . . . and alarming . . .

I but in the recent US electoral campaign,no-one wanted to talkabout climate change. It’s a hot potato.


I Climate modelling involves mathematical challenges ofunprecedented complexity.

Today’s climate modelsI gather data at ' 109 different locations,

















































I and try to improve the equations

I to make more accurate long-term predictions.



















They use the world’s most powerful computers, and havebecome steadily more convincing . . .

and alarming . . .



















I but in the recent US electoral campaign,

no-one wanted to talkabout climate change. It’s a hot potato.









I but in the recent US electoral campaign,no-one wanted to talkabout climate change.

It’s a hot potato.










Mathematical challenges of climate change


I Climate change is a hot potato - in the current US electoralcampaign, no-one has wanted to talk about it.

I Understanding uncertainty presents a challenge formathematics and democracy

I But hardest of all:agreeing to do something about it!




I Understanding uncertainty presents a challenge formathematics

and democracy











I But hardest of all:

agreeing to do something about it!






Harder than climate science . . .

Game Theory gives insights into why negotiations fail.

Harder than climate science . . .

Game Theory gives insights into why negotiations fail.

Sample “game”: the prisoner’s dilemma

Two professors, T and L, commit a crime, and are arrested. Theyare interrogated separately. Each has 2 options:

so 4 outcomes are possible

What do they do?

L

quiet

Keepsquiet

Confesses

T

Confesses

Keeps


Two professors, T and L, commit a crime, and are arrested. Theyare interrogated separately. Each has 2 options:

so 4 outcomes are possible

What do they do?

L

quiet

Keepsquiet

Confesses

T

Confesses

Keeps


Two professors, T and L, commit a crime and are arrested. Theyare interrogated separately. Each has 2 options:

so 4 outcomes are possible – each with its jail term

What do they do?

L

quiet

Keepsquiet

T

Confesses

Confesses

10 years

10 years 2 years

9 years

0 years 2 years

9 years

0 years

Keeps




What do they do?

L

quiet

Keepsquiet

T

Confesses

Confesses

10 years

10 years 2 years

9 years

0 years 2 years

9 years

0 years

Keeps




What do they do?

0 years

quiet

Keepsquiet

T

Confesses

L

Confesses

10 years

10 years 2 years

9 years

0 years 2 years

9 years

Keeps

Nash Equilibrium

Whatever T does, L does better to confess, and vice versa.

In every game, each player has to choose a strategy;

these choices determine the “payoff” for each.

A set of strategies (one for each player) is a Nash Equilibrium

if once they are adopted, no player can raise his payoff

by changing only his own strategy.

So a Nash equilibrium is best for everyone? Not necessarily!

Nash Equilibrium








Nash Equilibrium








Nash Equilibrium








Nash Equilibrium








Nash Equilibrium








Nash Equilibrium







So a Nash equilibrium is best for everyone?

Not necessarily!

Nash Equilibrium








In the prisoner’s dilemma,

T: confesses L: confesses

is a Nash equilibrium.

0 years

quiet

Keepsquiet

T

Confesses

L

Confesses

10 years

10 years 2 years

9 years

0 years 2 years

9 years

Keeps

They would have done better to keep quiet!




0 years

quiet

Keepsquiet

T

Confesses

L

Confesses

10 years

10 years 2 years

9 years

0 years 2 years

9 years

Keeps





0 years

quiet

Keepsquiet

T

Confesses

L

Confesses

10 years

10 years 2 years

9 years

0 years 2 years

9 years

Keeps


What’s the point of a game?

Like all mathematics, Game Theory takes complicated situationsand abstracts:

simplifies, throws away detail, . . . to revealunderlying structures. We can then see the same structuresappearing in many different contexts.

(Final abstraction in Prisoner’s Dilemma: instead of “keeps quiet”or “confesses”,“cooperates” or “defects”. More generallyapplicable.)

Example: Two companies compete, selling the same product. Ifthey cooperate, they can both sell at a high price and make biggerprofits. But if one defects by undercutting the other, he will sellmore, and his competitor will lose out.

The option of cooperating is called “forming a cartel” in thiscontext, and legislated against. Companies may not communicatetheir pricing intentions. This ensures that the incentives operate inexactly the same way as in the prisoner’s dliemma.


Like all mathematics, Game Theory takes complicated situationsand abstracts: simplifies, throws away detail, . . .

to revealunderlying structures. We can then see the same structuresappearing in many different contexts.





Like all mathematics, Game Theory takes complicated situationsand abstracts: simplifies, throws away detail, . . . to revealunderlying structures.

We can then see the same structuresappearing in many different contexts.





Like all mathematics, Game Theory takes complicated situationsand abstracts: simplifies, throws away detail, . . . to revealunderlying structures. We can then see the same structuresappearing in many different contexts.






(Final abstraction in Prisoner’s Dilemma: instead of “keeps quiet”or “confesses”,

“cooperates” or “defects”. More generallyapplicable.)





(Final abstraction in Prisoner’s Dilemma: instead of “keeps quiet”or “confesses”,“cooperates” or “defects”.

More generallyapplicable.)











Example: Two companies compete, selling the same product.

Ifthey cooperate, they can both sell at a high price and make biggerprofits. But if one defects by undercutting the other, he will sellmore, and his competitor will lose out.





Example: Two companies compete, selling the same product. Ifthey cooperate, they can both sell at a high price and make biggerprofits.

But if one defects by undercutting the other, he will sellmore, and his competitor will lose out.












Healthy competition ' Prisoner’s Dilemma

£200,000

T plc

L plc

Undercuts Fixes price

Fixes price

Undercuts

£1,500,000

£1,000,000

£200,000

£500,000

£500,000

£1,000,000£1,500,000

Nash equilibria

Not all games have a Nash equilibium. Some have more than one.

In the Stag Hunt (=“coordination game” or “trust dilemma”), twohunters collaborate hunting a stag, or each can hunt a hare alone.

1

Stag Hare

Hare

Stag

Sir L

Sir T

10

02

1

2 1

Nash equilibria

Not all games have a Nash equilibium. Some have more than one.In the Stag Hunt

(=“coordination game” or “trust dilemma”), twohunters collaborate hunting a stag, or each can hunt a hare alone.

1

Stag Hare

Hare

Stag

Sir L

Sir T

10

02

1

2 1

Nash equilibria

Not all games have a Nash equilibium. Some have more than one.In the Stag Hunt (=“coordination game” or “trust dilemma”), twohunters collaborate hunting a stag, or each can hunt a hare alone.

1

Stag Hare

Hare

Stag

Sir L

Sir T

10

02

1

2 1

Nash equilibriaNot all games have a Nash equilibium. Some have more than one.In the Stag Hunt (=“coordination game” or “trust dilemma”), twohunters collaborate hunting a stag, or each can hunt a hare alone.

1

Stag Hare

Hare

Stag

Sir L

Sir T

10

02

1

2 1

Here there are two Nash equilibria.

Nash equilibriaNot all games have a Nash equilibium. Some have more than one.In the Stag Hunt (=“coordination game” or “trust dilemma”), twohunters collaborate hunting a stag, or each can hunt a hare alone.

1

Stag Hare

Hare

Stag

Sir L

Sir T

10

02

1

2 1

Here there are two Nash equilibria.

Theorem: Every competitive game has a Nash equilibrium, ifmixed strategies are allowed.

A mixed strategy plays each strategy Si with probability pi .

The payoff is now the expected value∑i

pi Payoff (Si ).

This theorem and other related work (mathematically his mosttrivial) earned him the Nobel prize for Economics in 1994.

Its value had been tested for 40 years.




pi Payoff (Si ).





The payoff is now the expected value

∑i

pi Payoff (Si ).






pi Payoff (Si ).






pi Payoff (Si ).



How do players arrive at a Nash equilibrium?

Not necessarily through rational appraisal of the availablestrategies. Players in repeated games may reach a Nashequilibrium through trial and error – and it may be hard to escapefrom.

Evolutionary Game Theory studies how this occurs – and how‘players’ (societies, species, political parties, competingcompanies . . .) sometimes avoid falling into a damaging Nashequilibrium. How does altruism evolve?


Not necessarily through rational appraisal of the availablestrategies.

Players in repeated games may reach a Nashequilibrium through trial and error – and it may be hard to escapefrom.



Not necessarily through rational appraisal of the availablestrategies. Players in repeated games may reach a Nashequilibrium through trial and error –

and it may be hard to escapefrom.







Evolutionary Game Theory studies how this occurs – and how‘players’ (societies, species, political parties, competingcompanies . . .) sometimes avoid falling into a damaging Nashequilibrium.

How does altruism evolve?




The metaphor of a fitness landscape can help.

(Careful: physicists use energy landscapes, in which objectsnaturally seek out the lowest points)

The metaphor of a fitness landscape can help.

(Careful: physicists use energy landscapes, in which objectsnaturally seek out the lowest points)

Sub-optimal Nash equilibria in Public Goods games

Key idea: players reap benefits of their actions for themselves, butshare the costs among many . . .

. . . Privatise the gain, share the pain. . . .

1. “Tragedy of the Commons” Villagers graze their animals on

shared common land. If I graze my animals,

Benefit (all to me): My animals grow fat

Cost (shared among all): Grassland is depleted

Outcome if we all do it: Overgrazing and loss of shared resource





















shared common land.

If I graze my animals,





































2. Downward wages spiral

Companies compete by lowering the wages of their employees

Benefit: I reduce my production costs ⇒ I charge

lower prices ⇒ I gain higher market share and more profits

Costs : My workers’ purchasing power is reduced;

the resulting loss of sales is shared among all companies

Outcome: All companies do same ⇒ economic downturn




















Benefit:

I reduce my production costs ⇒ I charge








Benefit: I reduce my production costs

⇒ I charge









lower prices

⇒ I gain higher market share and more profits

















Costs :

My workers’ purchasing power is reduced;




























3. The diner’s dilemma, or, How we ate the world

Six friends dine together in a restaurant, agreeing to share the bill

equally, but making their choices individually. Two meal options:

Quite expensive meal mexpense e

pleasure p

Very expensive meal Mexpense E

pleasure P

Which to choose?


3. The diner’s dilemma,

or, How we ate the world




pleasure p


pleasure P

Which to choose?






pleasure p


pleasure P

Which to choose?




equally, but making their choices individually.

Two meal options:


pleasure p


pleasure P

Which to choose?






pleasure p


pleasure P

Which to choose?






pleasure p


pleasure P

Which to choose?






pleasure p


pleasure P

Which to choose?

Suppose E > P > p > e

if alone, I would choose m.

In company, the extra cost, to me, if I choose M is

E − e

6

So ifP > e + (E − e)/6

I choose M.

Increased benefit (all to me):

P − p

Increased cost (shared among all six diners):

E − e

We all choose M.

We all choose a meal that we would have preferred not to eat!




E − e

6

So ifP > e + (E − e)/6

I choose M.


P − p


E − e

We all choose M.





E − e

6

So ifP > e + (E − e)/6

I choose M.


P − p


E − e

We all choose M.





E − e

6

So ifP > e + (E − e)/6

I choose M.


P − p


E − e

We all choose M.





E − e

6

So ifP > e + (E − e)/6

I choose M.


P − p


E − e

We all choose M.





E − e

6

So ifP > e + (E − e)/6

I choose M.


P − p


E − e

We all choose M.





E − e

6

So ifP > e + (E − e)/6

I choose M.


P − p


E − e

We all choose M.





E − e

6

So ifP > e + (E − e)/6

I choose M.


P − p


E − e

We all choose M.





E − e

6

So ifP > e + (E − e)/6

I choose M.


P − p


E − e

We all choose M.





E − e

6

So ifP > e + (E − e)/6

I choose M.


P − p


E − e

We all choose M.


Stupid! But if I don’t have the expensive meal,

everyone else will,and I will still have to pay. So I opt for M.

In a study by Gneezy et al, groups of six subjects who went to arestaurant in response to an advertisement were offered meals,ostensibly in recompense for answering a questionnaire. Differentgroups had different modalities of payment:

1. Pay your own bill

2. Split the bill

3. Pay one sixth of the bill

4. Pay nothing.

Consumption: 4 > 3 > 2 > 1.

Now imagine there are 7,192,824,796 diners, splitting the bill.

There are!

Fortunately, not everyone attends climate negotiations.

Stupid! But if I don’t have the expensive meal, everyone else will,

and I will still have to pay. So I opt for M.



2. Split the bill


4. Pay nothing.



There are!


Stupid! But if I don’t have the expensive meal, everyone else will,and I will still have to pay.

So I opt for M.



2. Split the bill


4. Pay nothing.



There are!


Stupid! But if I don’t have the expensive meal, everyone else will,and I will still have to pay. So I opt for M.



2. Split the bill


4. Pay nothing.



There are!



In a study by Gneezy et al,

groups of six subjects who went to arestaurant in response to an advertisement were offered meals,ostensibly in recompense for answering a questionnaire. Differentgroups had different modalities of payment:


2. Split the bill


4. Pay nothing.



There are!



In a study by Gneezy et al, groups of six subjects who went to arestaurant in response to an advertisement were offered meals,ostensibly in recompense for answering a questionnaire.

Differentgroups had different modalities of payment:


2. Split the bill


4. Pay nothing.



There are!





2. Split the bill


4. Pay nothing.



There are!





2. Split the bill


4. Pay nothing.



There are!





2. Split the bill


4. Pay nothing.



There are!





2. Split the bill


4. Pay nothing.



There are!





2. Split the bill


4. Pay nothing.



There are!


Still, more than 100 countries participate.

The countries negotiating are like the diners:

if I don’t continue burning fossil fuels, everyone else will, and I will

still suffer the effects of climate change, and worse still, with a

weaker economy due to my reduced use of fossil fuels. So no treaty

is agreed. This inaction is a malign Nash equilibrium.









if I don’t continue burning fossil fuels,

everyone else will, and I will






if I don’t continue burning fossil fuels, everyone else will,

and I will







still suffer the effects of climate change, and worse still,

with a







weaker economy due to my reduced use of fossil fuels.

So no treaty







is agreed.

This inaction is a malign Nash equilibrium.







How to leave a sub-optimal Nash Equilibrium ?

Alliances change the nature of the game.

I In Prisoner’s dilemma, if the two accused are able tocoordinate their strategy, they both choose to keep quiet.Companies (secretly and illegally) form cartels.

I If villagers agree to limit grazing, a tragedy of the Commons

can be averted. This happens naturally in communities whichare small enough for everyone to know everyone else: thesocial stigma attached to over-use of the shared resource canbe sufficient disincentive.

I A trades union, or a minimum wage, which ensures wages are

not depressed, can avoid a downward wages spiral.

I Almost all cooperative action requires sanctions againstfreeloaders.











I In Prisoner’s dilemma, if the two accused are able tocoordinate their strategy,

they both choose to keep quiet.Companies (secretly and illegally) form cartels.








I In Prisoner’s dilemma, if the two accused are able tocoordinate their strategy, they both choose to keep quiet.

Companies (secretly and illegally) form cartels.






































Understanding that a Nash equilibrium is not necessarily optimal

can help negotiators seek alliances.

Members of parliament responsible for later ratification of treatiesalso need this understanding.

And voters have to support them.









“The tragedy of the commons”

(Garret Hardin, Nature, 1968).

“We can make little progress in working toward optimumpopulation size until we explicitly exorcise the spirit of Adam Smithin the field of practical demography. In economic affairs, TheWealth of Nations (1776) popularized the invisible hand, the ideathat an individual who intends only his own gain, is, as it were, ledby an invisible hand to promote the public interest. Adam Smithdid not assert that this was invariably true, and perhaps neither didany of his followers. But he contributed to a dominant tendency ofthought that has ever since interfered with positive action based onrational analysis, namely, the tendency to assume that decisionsreached individually will, in fact, be the best decisions for an entiresociety. If this assumption is correct it justifies the continuance ofour present policy of laissez faire in reproduction. If it is correct wecan assume that men will control their individual fecundity so as toproduce the optimum population. If the assumption is not correct,we need to reexamine our individual freedoms to see which onesare defensible.”

“The tragedy of the commons”(Garret Hardin, Nature, 1968).


“The tragedy of the commons”(Garret Hardin, Nature, 1968).


References

Ken Binmore, A very short introduction to Game Theory, OxfordUniversity Press 2007

Garrett Hardin, The Tragedy of the Commons, Science,162(1968):1243-1248.

Uri Gneezy, Ernan Haruvy, Hadas Yafe, The inefficiency of splittingthe bill, The Economic Journal, 114 (April 2004) , 265-280

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