Us And Them, a Computational Game Theory Analysis of Competition and Cooperation

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Us And Them, a Computational Game Theory
Analysisof Competition and Cooperation
Lance Hanlen
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Background
Computer Science and Game Theory, a
common ancestry.
A common family business dynamical systems
A striking family resemblance agents and players
Just a sprig of good ol prairie-style
inbreeding Computational Game Theory. Lots of
one-sided conversations some quite good!
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Game Theory
Prisoners Dilemma
Prisoners Dilemma
Prisoners Dilemma
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Nash equilibrium
Wikipedia A stable state of a system that
involves several interacting participants in
which no participant can gain by a change of
strategy as long as all the other participants
remain unchanged.
Nash Equilibrium for Prisoners dilemma (fink,
fink)
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Relative Preferences
Evidence exists that under some circumstances
people behave altruistically, that is, they put a
positive value on the welfare of others.
Alternatively, people are known to behave
enviously (for lack of a better term), that is,
they put a negative value on the welfare of
others.
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Relative Preferences
A series of papers, mostly from quantitative
biology or economics, have put forth evolutionary
arguments trying to explain one or the other of
these
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Evolutionarily Stable Parameter
Originally due to John Maynard Smith.
Many variations, we use one thats very close to
Smiths original notion.
No easy way to say it!
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Evolutionarily Stable Parameter
Wikipedia An ESS depends on the idea of
invasion, where a population of strategy-X
players is visited by a strategy-Y player. The
new player is said to invade if, following
strategy Y, he scores better than the average
strategy-X player. Assuming players are able to
choose and switch strategies, this would induce
the indigenous population to start switching to
strategy Y. In many cases there are diminishing
returns for the later adopters, and what follows
is an equilibrium ratio of strategy-X players to
strategy-Y players.
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Evolutionarily Stable Parameter
A strategy X is evolutionarily stable if there
is no strategy Y that can invade it. That is,
anybody bringing a new strategy into a population
of strategy-X players will fare no better on
average than the X players are already doing.
(See the closely-related Nash equilibrium)ESS is
stable in respect to randomly and occasionally
occurring invading strategies, thus it is not
stable in respect to mass counts of invaders.
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Altruism
In 1998 Bester and Guth considered some topics
economists, as a rule, arent comfortable talking
about

• People who risk their lives for others
• Soldiers voluntarily going to war
• The many forms of charity

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Altruism
They propose a simple game played between player
1 and player 2
Player 1s action x
Player 2s action y
Denote the valuation that each player attaches to
the games outcome Utility
Utility is the material payoff or evolutionary
success (fitness)
In this game, define Utility for each player as
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Altruism
The higher the (model parameter representing)
altruism, the closer the (model parameters
representing) the respective players fitness to
their highest possible value.
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Envy
In 1999 Possajennikov applied the model inversely
and was able to show analytically that it also
worked to explain (the dynamical behavior of a
model inspired by) spite.
More recently, Eaton and Eswaran extended the
model dramatically to study entire populations of
competing players in a (dynamical system)
environment modeled to resemble hunter gatherer
societies in the early stone age.
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Envy
Fitness levels of a typical incumbent and that of
the mutant as a function of the mutants
preference parameter, µ, when all incumbents
have their preference parameter fixed at µ(n).
The mutants fitness naturally equals that of an
incumbent when µ µ(n), but for all other
values of µthe mutants fitness is lower.
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Envy
To complete the characterization of the
evolutionarily stable equilibrium, we invoke the
condition that fitness must be equal to
subsistence in equilibrium. the preference
parameter becomes more and more negative while
the population size heads towards unity
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Envy
Continuing in this manner we can analyze the
model to predict population densities for hunter
gatherer societies in the stone age.
In summary, the model predicts sparse
populations, and is very specific in detailed
explanation.
The models predictions are borne out by
unrelated anthropological studies of both recent
and Neolithic hunter gatherer society population
dynamics.
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Envy
Clearly a powerful model, but strictly analytical
(40 pages of derivations)
What do we gain by simulating with evolutionary
algorithms?
At least the answer to that question, and thats
a lot!
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Social Dilemma
Prisoners dilemma in a continuous space.
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Social Dilemma

• Less is more Plain Substitutes.
• Even more still is even worse Strategic
  Substitutes.

• FIRST MOVER ADVANTAGE
• IF BOTH MOVE, BOTH END UP WORSE OFF

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Social Dilemma
Player 1 costlessly shifts RF1 up.

• Common property fishery pre-commitment to
  technically superior, but uneconomic fishing gear
  that makes one a more aggressive competitor
• Cournot oligopoly for firm 1 to produce a
  carefully chosen larger quantity before firm 2
  chooses its quantity, forcing an equilibrium at
  the intersection of 1s pre-committed quantity
  and 2s reaction function this is, of course,
  the Stackelberg (1934) idea
• strategic inventory accumulation
• pre-committing productive inputs so as to reduce
  variable costs thus shifting the firms reaction
  function

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Social Dilemma

• A government whose objective is to maximize
  welfare has an incentive to provide an export
  subsidy to its domestic firm if the other
  government does nothing, the optimal export
  subsidy shifts the domestic firms reaction
  function so that it intersects the foreign firms
  reaction function at the Stackelberg equilibrium.
  When both countries choose to subsidize exports,
  equilibrium welfare declines in both countries
  from Brander and Spencer (1985)

Player 1 costlessly shifts RF2 down.

• acquisition and hoarding of superior productive
  inputs, thereby raising the other firms costs
  and shifting its reaction function in the desired
  manner

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Social Dilemma
Players have an incentive to explore
pre-commitments that either make them more
aggressive competitors or make their competitors
less aggressive when pre-commitment is possible
and attractive there is a first mover advantage
when both players pre-commit they both end up
worse off than they are in the absence of
pre-commitment.
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Social Dilemma

• More is better Plain Compliments.
• Even more still is even better yet Strategic
  Complements.

• SECOND MOVER ADVANTAGE
• IF BOTH MOVE, BOTH END UP BETTER OFF

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Social Dilemma
Player 1 costlessly shifts RF1 up (over).

• Bertrand duopoly in which price setting firms
  sell goods that are imperfect substitutes
• as a Cournot duopoly in which quantity setting
  firms sell goods that are complements
• as a model of production with a positive
  externality

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Social Dilemma

• More is better Plain Complements.
• Even more still not as good Strategic
  Substitutes.

• FIRST MOVER ADVANTAGE
• IF BOTH MOVE, BOTH END UP WORSE OFF

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Social Dilemma
Allied countries (like the U. S. and Canada), xi
as country is expenditure on homeland security
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Social Dilemma

• Less is more Plain Substitutes.
• Even more still is not as bad Strategic
  Complements.

• SECOND MOVER ADVANTAGE
• IF BOTH MOVE, BOTH END UP BETTER OFF

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Social Dilemma
Enemy countries (like the U. S. and Canada), xi
as country is expenditure on homeland security
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Social Dilemma
Multiple players
Eaton Eswaran model structure
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Social Dilemma
Applying a preference parameter
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Social Dilemma
Equilibrium is much more efficient in the
presence of an altruistic preference in each
players strategy
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Social Dilemma
Equilibrium is much more efficient in the
presence of an envious preference in each
players strategy
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Social Dilemma
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Social Dilemma
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Social Dilemma The Final Frontier
Combined model, very difficult and uncharted
territory.
We already know there is no analytical way.
We will go where no game theorist has gone
before, evolutionary algorithms.
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Social Dilemma The Final Frontier
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Social Dilemma The Final Frontier
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References
Bester, H. and W. Guth (1998), Is Altruism
Evolutionarily Stable?, Journal of Economic
Behaviour and Organization, 34, pp. 193-209.
Brander, J.A., and B.J. Spencer Export Subsidies
and International Market Share Rivalry. Journal
of International Economics 18, 83100
C.B. Eaton. The Elementary Economics of Social
Dilemmas Presidential address to the Canadian
Economics Association, 2004.
C.B. Eaton, and M. Eswaran. The Evolution of
Competition The Canadian Journal of Economics,
2003.
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References
Possajennikov, A. (2000), On the Evolutionary
Stability of Altruistic and Spiteful
Preferences, Journal of Economic Behaviour and
Organization, 42, pp. 125-129.
John Maynard Smith and George R. Price (1973).
"The logic of animal conflict." Nature
Wikipedia April, 2005 http//www.absoluteastronomy
.com/encyclopedia/e/ev/evolutionarily_stable_strat
egy.htm