Download Dardi on game theory

Acqui Terme, 2 September 2010
Cooperation from a Game
Theory perspective
Marco Dardi
University of Florence
[email protected]
In common language cooperation means mutual
assistance in order to get benefits that would be
unavailable to agents acting non-cooperatively. GT
can be applied to analyzing problems that arise in the
effort to reap this cooperative surplus.
In the language of GT the terms cooperative/noncooperative have also a technical meaning connected
with the formal specification of solution concepts.
This lecture investigates the relationship between the
two levels at which GT deals with cooperation: as an
object problem and as a formal definition.
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1. GT, conflict, cooperation
In a nutshell: GT elaborates methods for
•
describing the structure of interactive situations, focusing
on the choices available to individual agents, on their
knowledge and preferences (theory of representations)
•
prescribing ways of behaving in interactive situations
that comply with notions of intelligent pursuit of each
agent’s own interests (theory of solutions)
First, representation. The normal or strategic form of socalled non-cooperative games (on these, later on) is the
most convenient representation for the purposes of this
lecture. It constitutes a sort of compromise between the
extensive form representation, that can be reduced to
normal form at the cost of losing relevant detail, and the
coalitional form representation of so-called cooperative
games, that can be derived from the normal form, again with
some loss of detail.
The normal form representation consists of:




a set of agents or players (in number n  1)
for each agent, a set of strategies  Strategic space
a set of outcomes
for each agent, preferences on the outcomes 
represented by von Neumann utility numbers
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According to the normal form representation, a game form
G is a mapping
G: strategic space  outcomes
A game g is a game form + a specification of the
preferences of the agents involved. Formally, with a
numerical representation of preferences, a game is a
composite mapping g = U(n) • G, with U(n) the n-vector of the
players’ utility functions. Hence,
g: strategic space  n
Interpretation: G describes the “physical” rules that apply in
a situation (distribution of the agents’ powers over the
outcomes). Each G generates a class of games, one for
each specification of the preferences of the agents involved
(think of a plan of organization and all the firms organized
according to it; of a legal code, and all the specific situations
regulated by it; etc)
Warning: the main behavioral feature implicit in GT
representations is not self-interest or egoism, but the
instrumentality of the choices made in playing a game.
Players are not interested in the actions they or the other
players choose, but in the outcomes brought about by these
actions. The motivations lying behind their preferences on
the outcomes needn’t be made explicit. Altruism is as good
a motivation as any other.
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Just looking at the images of g in n we get an idea of what
conflict and cooperation look like in a GT framework.
Case 1: the vector inequality  (strong Pareto preference)
does not apply in the range of g  all the outcomes
are Pareto optima  g describes a pure conflict
situation
Case 2: The range of g is completely ordered by  
Pareto and individual preferences are in agreement 
conflict is non-existent
Case 3: The range of g is ordered by  but only partially 
some (more than one) outcomes are Pareto optima,
some are Pareto-dominated  conflict and common
interest co-exist
In 1 there are no gains from cooperation (e.g., zero-sum
games). In 2, the gains can be reaped unproblematically
because individual and common interests accord with each
other (the only problem may be coordination). The only
case of interest for GT turns out to be 3, where conflicts of
interest may prevent the players from picking the potential
gains from cooperation.
Note: the conflict/cooperation mix does not depend on the
game form but on “who” the players are
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2. Cooperation problems: basic patterns
To get an idea, take the simplest possible case: two agents,
A and B, each with two strategies, c = follow a cooperative
line of conduct, d = don’t follow such a line (whatever this
may mean). No player can make his/her choice conditional
upon the other player’s choice (no information leaks).
Game form (outcomes in greek letters)
B
c
d
c


d


A
Two games (
= A’s preference;
= B’s preference)
B
c
A
d
B
c
d



c
d
c


d


A

“PRISONER”
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“HAWK-DOVE”
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In both games we can observe: gains from cooperation,
symmetrical incentives to defect, each player prefers the
other to cooperate. PRISONER unconditionally prefers not
to cooperate; H-D prefers to cooperate if the other doesn’t.
Consequently, gains from cooperation are greater in H-D
than PRISONER as revealed by Pareto preference
B
c
A
d
B
c
d




PRISONER
c
d
c


d


A
HAWK-DOVE
In order to avoid the non-cooperative, Pareto-dominated
outcome  some sort of agreement is needed. Any
agreement in PRISONER is liable to defection. So is
agreement  in H-D, while agreements  and , although
secure against defection, may be refused on grounds of
justice.
The analysis suggests remedies that in all cases, barring the
possibility of changing the agents’ preferences, require
modifications of the game form…
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In PRISONER: expand the game so as to add a post-play
stage in which players have a possibility of sanctioning the
agreement by means of penalties. The expansion may
consist of a number of repetitions of the game, provided no
repetition is known with certainty to be the last one.
In H-D: introduce correlated randomization (NB: not the
independent randomization known as “mixed strategies”) of
the outcomes. In some cases this will require changes of the
game form through the introduction of an umpire or an
external information system (with suitable utility numbers the
best fair agreement requires prob() = 0 and  to be drawn
with the same probability as  and ).
These examples provide a clue to one of the most thriving
lines of research in applied GT during the 1980s and 1990s:
how to design a game form such that the cooperative gains
latent in a situation do not go unexploited (implementation
theory (IT) or “mechanism design”). IT provides a new basis
for the theory of contracts, industrial organization, imperfect
markets and other microeconomic applications.
The object of IT is: given the agents’ preferences, re-design
the situation as a game such that sticking to the agreement
to cooperate in reaping the existing Pareto gains is the only
intelligent line of conduct for all the players in the game. A
preliminary step is of course clarifying what an intelligent line
of conduct in a game is. This leads to the particular theory of
solutions on which IT depends…
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3. Stability
In the theory of solutions a solution concept is a rule that
selects a subset of strategy profiles out of the strategic
space of the game according to some criterion of
rationality,
solution  strategic space
with solution being defined by some common property of
the strategy profiles included.
The main property used in GT solution concepts is that of
stability. A strategy profile is stable whenever for each
player the following statement holds: “if nobody has any
reason to refuse to do his/her part in this profile, I have no
reason either”.
A cooperative agreement has a chance of being effective
only if it prescribes strategies which make up a profile that
belongs to a stable solution. Thus, cooperative
agreements should be stable solutions of an appropriately
designed game.
Stability may mean various things depending on the way
“reasons to refuse” in the above statement are specified.
Here for the first time we have to deal with the technical
distinction in GT between so-called non-cooperative and
cooperative solutions.
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From now on: cooperative/non-cooperative in the technical
sense of GT solutions will be marked off with a *.
A stable solution is non-cooperative* if “reasons to refuse”
are referred exclusively to individual players. An individual
player has reason to refuse to do his/her part in a prescribed
strategy profile if, in the hypothesis that the others do theirs,
he/she has the power of bringing about a preferred outcome.
If no player has such reasons, then the prescribed strategy
profile is a Nash non-cooperative* equilibrium (NE). Hence,
a stable non-cooperative* solution coincides with the set of
Nash non-cooperative* equilibria.
Note: by basing cooperative agreements on NE, as is
usually done in IT, we have a theory of cooperation in which
the stability of agreements relies entirely on a noncooperative* solution concept. Far from being a paradox,
this is the essence of the so-called “Nash program” (Nash
1951) for reducing all cooperative solutions to noncooperative* equilibrium analysis. But it is to be noted that
Von Neumann and Morgenstern refused to take Nash’s
cooperative*/non-cooperative* partition into consideration.
The statement
“the general, typical game – in particular all significant problems of
a social exchange economy – cannot be treated without these
devices [of cooperation]”
remained unchanged in the third edition of their work (1953,
p. 44), after they had taken cognizance of Nash’s papers on
non-cooperative* games.
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Stable cooperative* solutions differ from NE in that “reasons
to refuse” may be referred not only to individuals but also to
groups of individuals acting cooperatively (coalitions).
Refusing is not necessarily an individual affair, individuals
may refuse by forming a coalition in order to get a better
outcome.
A coalition has reason to refuse to do its part in a prescribed
strategy profile if, in the hypothesis that the other players do
theirs, it has the power of bringing about an outcome which
is preferred by all its members. That a strategy profile
belongs to NE is a necessary but not a sufficient condition
for it to be stable with respect to coalitions. Generally,
cooperation in refusing restricts the domain of stability.
If all possible coalitions are considered to be equally
feasible, the relevant cooperative* solution concept is (from
Edgeworth) the CORE. A strategy profile belongs to the
CORE if and only if no individual or coalition has reason to
refuse to do its part in it on condition that all the others do
theirs.
Obviously, CORE  NE. Basing a cooperative agreement
upon a cooperative* solution concept turns out to be more
difficult than relying on a non-cooperative* solution such as
NE. In particular cases the CORE may even be empty (for
example, PRISONER has NE = (d,d) but no CORE).
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However, not all the conceivable coalitions are generally
equally feasible. More sophisticated concepts of
cooperative* stability refer to the stability of the coalitions
themselves. A strategy profile s, which is liable to be
refused by a coalition, may be considered to be stable all
the same if the agreement within the potentially refusing
coalition can in turn be challenged by some other coalition.
Members of the former, knowing that the latter could thwart
their plans, could be dissuaded from forming it. Thus,
although not in the CORE, s may remain unchallenged.
These considerations open the way to a variety of
sophisticated cooperative* solution concepts (BARGAINING
SET, KERNEL, SHAPLEY VALUE etc.). The specific
definitions depend on the way that the notion of challenging
coalitions and counter-coalitions is modelled. In general, all
these concepts are more permissive than the CORE.
Von Neumann & Morgenstern proposed a cooperative*
solution concept that, while of a sophisticated kind, lies on a
completely different line from those we have considered so
far. Their concept of “stable set” (SSET) was defined not on
the basis of a common property of the strategy profiles that
belong to it, but on the basis of a structural property of the
set itself. They insisted on stability in social theories being
“a property of the system as a whole and not of the single
imputations [here, read “strategy profiles”] of which it is
composed” (1953, p. 36).
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A set of strategy profiles is a SSET if and only if
(i) For each profile included in it: if a coalition has reason
to refuse it, this must be in favour of a profile excluded;
(ii) For each profile excluded from it: there is at least a
coalition that has reason to refuse it in favour of a
profile included
SSET may be empty (as in PRISONER). In the same game
there may be more than one SSET (as in the purely
conflictual game known as “matching pennies”, with empty
NE and CORE ). If more than one SSET exist, these have
no intersection. And, of course, in general CORE  SSET,
while there is no general relationship between SSET and
NE.
Von Neumann & Morgenstern interpreted SSET as a
formalization of the notion of an “established order of
society” or “accepted standard of behavior”. It describes a
variety of modes of behavior, none of which is able to
unsettle the others. Some of them may be unsettled by
some non-conforming modes of behavior, but all of the
latter are unsettled by one or another of the accepted
ones. Lastly, the same game or social situation may
express more than one such “order” or standard. However,
for all its evocative power, this notion has had little
application in economics and in social theory in general.
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4. Final remarks
Cooperative* solution concepts have been little used in
economic applications. Perhaps the very variety of
concepts available, with the ensuing feeling of ad hoc
constructions, has been an obstacle to generalized
adoption. Thus, cooperation on a non-cooperative* basis,
in the sense explained above, constitutes the unifying
methodological framework of great part of contemporary
microeconomics (Moulin, 1995: “Cooperation in the
economic tradition is mutual assistance between egoists”).
The “Nash program” has prevailed over Von Neumann &
Morgenstern’s more “social” approach.
But note that Moulin’ s reference to “egoism”, as remarked
above (slide 4), is anyway wide of the mark, since acting
on the basis of individual preferences has no necessarily
egoistic implications.
Moreover, we should not be induced to view the noncooperative* approach to cooperation as an expression of
an inevitably atomistic social philosophy. All the
arguments that try to justify NE as the only solution
concept consistent with individual rationality resort to
some kind of “communality of thought” that, as Schelling
(1960) and Lewis (1969) have pointed out, presupposes
that some social convention is in force. It cannot be an
exclusively individual affair.
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Recall the premise of the conditional statement underlying
individual stability (slide 9): “if the others have no reason
to refuse to do their part in this profile, I have no reason
either”. Doing my part is rational if the premise is true. But
why should I believe it to be true? A moment’s reflection
shows that the basis of this belief is the belief that it is
shared by everybody, that it is believed to be shared by
everybody, and so on ad infinitum: briefly, it must be, in a
specially strong sense, a “common” belief.
Common beliefs presuppose conscience that on some
matters there is something like communality of thought,
thought that does not need to be communicated. This
would seem to be an unlikely phenomenon in a rigorously
atomistic society because it implies that individuals do not
think independently of each other on all matters, and
therefore some “accepted standards” of thought (in Von
Neumann & Morgenstern’s language) must be wellestablished. Individualism itself must be an expression of
such a standard.
Thus, non-cooperative* foundations of cooperation (as in
IT) must in turn be founded on sociological premises lying
at a deeper level than individual rationality. Nash program
is at best half a program for GT-based social research; the
other half would require relating solution concepts to types
of social culture.
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5. Selected references
For the basics of GT, IT, coalitions: according to my taste
and teaching experience, the best (although by no means
elementary) introduction is Osborne & Rubinstein, A
course in GT (1994) MIT Press, chapters 1-5, 10, 13-14.
For the game-theoretic outlook on cooperation in
economics: see Moulin, Cooperative microeconomics
(1995) Prentice Hall.
For stability, cooperative/non-cooperative games, and the
Nash program: see von Neumann & Morgenstern, Theory
of games and economic behavior (3rd edition, 1953)
Princeton UP, chapts. I.4, V, XI; Nash, “Noncooperative
games”, Annals of Mathematics (1951); Myerson, “Nash
equilibrium and the history of economic theory”, JEL
(1999).
For the foundations of non-cooperative NE: Schelling,
The strategy of conflict (1960) Harvard UP; Lewis,
Convention: A philosophical study (1969) Blackwell;
Bacharach, “A theory of rational decision in games”,
Erkenntnis (1987).
For all matters related to GT: see the relevant chapters in
the Handbook of GT with economic applications, Aumann
& Hart editors, 3 vols., 1992-1994-2002: North-Holland.
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