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Prisoners Dilemma |
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| Since in any situation playing defect is more beneficial than cooperating, all rational players will play defect, all things being equal.Cooperation may then arise as an equilibrium outcome.The incentive to defect is overcome by the threat of punishment, leading to the possibility of a cooperative outcome.So if the game is infinitely repeated, cooperation may be a subgame perfect Nash equilibrium although both players defecting always remains an equilibrium and there are many other equilibrium outcomes.Tucker formalized the game with prison sentence payoffs and gave it the "Prisoner's Dilemma" name (Poundstone, 1992).The classical prisoner's dilemma (PD) is as follows:
Two suspects, A and B, are arrested by the police.Each prisoner must make the choice of whether to betray the other or to remain silent.The dilemma arises when one assumes that both prisoners only care about minimizing their own jail terms.If you knew the other prisoner would stay silent, your best move is to betray as you then walk free instead of receiving the minor sentence.If you knew the other prisoner would betray, your best move is still to betray, as you receive a lesser sentence than by silence.Yet by both defecting they get a lower payoff than they would get by staying silent.Even when they both know the other to be rational and selfish, they will both play defect.Defect is what they will play no matter what, even though they know fully well that the other player is playing defect as well and that they will both be better off with a different result.Generalized form
We can expose the skeleton of the game by stripping it of the prisoner framing device.The generalized form of the game has been used frequently in experimental economics.There are two players and a banker.Each player holds a set of two cards: one printed with the word "Cooperate", the other printed with "Defect" (the standard terminology for the game).If player 1 (red) defects and player 2 (blue) cooperates, player 1 gets the Temptation to Defect payoff of 5 points while player 2 receives the Sucker's payoff of 0 points.If both cooperate they get the Reward for Mutual Cooperation payoff of 3 points each, while if they both defect they get the Punishment for Mutual Defection payoff of 1 point.These point assignments are given arbitrarily for illustration.Let T stand for Temptation to defect, R for Reward for mutual cooperation, P for Punishment for mutual defection and S for Sucker's payoff.In addition to the above condition, if the game is repeatedly played by two players, the following condition should be added.Amongst results shown by Nobel Prize winner Robert Aumann in his 1959 paper, rational players repeatedly interacting for indefinitely long games can sustain the cooperative outcome.Popular interest in the iterated prisoners dilemma (IPD) was kindled by Robert Axelrod in his book The Evolution of Cooperation (1984).Axelrod invited academic colleagues all over the world to devise computer strategies to compete in an IPD tournament.The programs that were entered varied widely in algorithmic complexity; initial hostility; capacity for forgiveness; and so forth.He used this to show a possible mechanism for the evolution of altruistic behaviour from mechanisms that are initially purely selfish, by natural selection.The best deterministic strategy was found to be "Tit for Tat," which Anatol Rapoport developed and entered into the tournament.Depending on the situation, a slightly better strategy can be "Tit for Tat with forgiveness."Nice
The most important condition is that the strategy must be "nice", that is, it will not defect before its opponent does.Retaliating
However, Axelrod contended, the successful strategy must not be a blind optimist.Forgiving
Another quality of successful strategies is that they must be forgiving.Though they will retaliate, they will once again fall back to cooperating if the opponent does not continue to play defects.One of the most important conclusions of Axelrod's study of IPDs is that Nice guys can finish first.PD game is simply defection; as explained above, this is true whatever the composition of opponents may be.PD game the optimal strategy depends upon the strategies of likely opponents, and how they will react to defections and cooperations.That individual is at a slight disadvantage because of the loss on the first turn.Tat players, the optimal strategy for an individual depends on the percentage, and on the length of the game.Monte Carlo simulations of populations have been made, where individuals with low scores die off, and those with high scores reproduce (a genetic algorithm for finding an optimal strategy).The mix of algorithms in the final population generally depends on the mix in the initial population.Once this recognition was made, one program would always cooperate and the other would always defect, assuring the maximum number of points for the defector.Southampton player, it would continuously defect in an attempt to minimize the score of the competing program.Tat is certainly a better strategy.Because of this new rule, this competition also has little theoretical significance when analysing single agent strategies as compared to Axelrod's seminal tournament.If an iterated PD is going to be iterated exactly N times, for some known constant N, then it is always optimal to defect in all rounds.The only possible Nash equilibrium is to always defect.However, even in this case always defect is no longer a strictly dominant strategy, only a Nash equilibrium.Another odd case is "play forever" prisoner's dilemma.The prisoner's dilemma game is fundamental to certain theories of human cooperation and trust.In 1975, Grofman and Pool estimated the count of scholarly articles devoted to it at over 2,000.Learning psychology and game theory
Where game players can learn to estimate the likelihood of other players defecting, their own behaviour is influenced by their experience of the others' behaviour.Simple statistics show that inexperienced players are more likely to have had, overall, atypically good or bad interactions with other players.The early transactions experienced by immature players are likely to have a greater effect on their future playing than would such transactions affect mature players.This principle goes part way towards explaining why the formative experiences of young people are so influential and why they are particularly vulnerable to bullying, sometimes ending up as bullies themselves.If the group is small the positive behaviour is more likely to feed back in a mutually affirming way, encouraging individuals within that group to continue to cooperate.Such processes are major concerns within the study of reciprocal altruism, group selection, kin selection and moral philosophy.One resolution of the dilemma proposed by Douglas Hofstadter in his Metamagical Themas is to reject the definition of "rational" that led to the "rational" decision to defect.In this view, truly rational (or "superrational") players take into account that the other person is (presumably) superrational, like them, and thus they cooperate.It's most prudent to give up straightforward maximizing and instead adopt a disposition of constrained maximization, according to which one resolves to cooperate with all similarly disposed persons and defect on the rest.In other words, moral constraints are justified because they make us all better off, in terms of our preferences (whatever they may be).Those that defect can be predicted because people are not completely opaque.Douglas Hofstadter expresses a strong personal belief that the mathematical symmetry is reinforced by a moral symmetry, along the lines of the Kantian categorical imperative: defecting in the hope that the other player cooperates is morally indefensible.The prisoner's dilemma is therefore of interest to the social sciences such as economics, politics and sociology, as well as to the biological sciences such as ethology and evolutionary biology.Many natural processes have been abstracted into models in which living beings are engaged in endless games of Prisoner's Dilemma (PD).This wide applicability of the PD gives the game its substantial importance.Both will reason that they have two options, either to increase military expenditure or to make an agreement to reduce weapons.The paradox is that both states are acting rationally, but producing an apparently irrational result.In sociology or criminology, the PD may be applied to an actual dilemma facing two inmates.In program management and technology development, the PD applies to the relationship between the customer and the developer.In the end, this will likely lead to a victory for the second cyclist (defecting) who has an easy ride in the first cyclist's slipstream.Also in athletics, there is a widespread practice in high school wrestling where the participants intentionally lose unnaturally large amounts of weight so as to compete against lighter opponents.In doing so, the participants are clearly not at their top level of physical and athletic fitness and yet often end up competing against the same opponents anyway, who have also followed this practice (mutual defection).Yet if a participant maintains their natural weight (cooperating), they will most likely compete against a stronger opponent who has lost considerable weight.When cigarette advertising was legal in the United States, competing cigarette manufacturers had to decide how much money to spend on advertising.Likewise, the profit derived from advertising for Firm B is affected by the advertising conducted by Firm A.If both Firm A and Firm B chose to advertise during a given period the advertising cancels out, receipts remain constant, and expenses increase due to the cost of advertising.Both firms would benefit from a reduction in advertising.However, should Firm B choose not to advertise, Firm A could benefit greatly by advertising.Nevertheless, the optimal amount of advertising by one firm depends on how much advertising the other undertakes.For instance, cigarette manufacturers endorsed the creation of laws banning cigarette advertising, understanding that this would reduce costs and increase profits across the industry.This analysis is likely to be pertinent in many other business situations involving advertising.For any software that is under the GPL, it is illegal to distribute only the unmodifiable form, including any changes made, thus forcing cooperation.The collective reward for unanimous (or even frequent) defection is very low payoffs (representing the destruction of the "commons").The commons are not always exploited: William Poundstone, in a book about the Prisoner's Dilemma (see References below), describes a situation in New Zealand where newspaper boxes are left unlocked.Newspapers are less risky to distribute under the honour system than other consumables because taking more than one offers very little extra benefit.One of several examples he used was "closed bag exchange":
Two people meet and exchange closed bags, with the understanding that one of them contains money, and the other contains a purchase.And what about agents, who charge a fee for organising these bag exchanges?Game Show Network in the United States.On the game show, three pairs of people compete.As each pair is eliminated, they play a game of Prisoner's Dilemma to determine how their winnings are split.If one cooperates and the other defects (Foe), the defector gets all the winnings and the cooperator gets nothing.If you know your opponent is going to vote Foe, then your choice does not affect your winnings.In a certain sense, Friend or Foe has a payoff model between "Prisoner's Dilemma" and "Chicken".Preference, Belief, and Similarity: Selected Writings.When one is considering the game itself, communication has no effect whatsoever.Prisoner's Dilemma is not affected in any way by communications.An alternative way of putting it is using the Darinian ESS simulation.Quakers, who always dealt honourably with their business partners.Potentially, it might be used to explain Wikipedia contributions: Text may be added under the assumption that if contributions are not made, then similar people will also fail to contribute (i.The Evolution of Cooperation.Kenneth Binmore, Fun and Games.Prisoner's Dilemma Doubleday, NY NY.University of Michigan Press.Evolutionary dynamics of the continuous iterated Prisoner's Dilemma, Journal of Theoretical Biology Full text
A.Further reading
Plous, S.Journal of Peace Research, Vol.All text is available under the terms of the GNU Free Documentation License.See Copyrights for details.Tanya and Cinque have been arrested for robbing the Hibernia Savings
Bank and placed in separate isolation cells.Both care much more about
their personal freedom than about the welfare of their accomplice.If you confess and your accomplice
remains silent I will drop all charges against you and use your
testimony to ensure that your accomplice does serious time.If you both confess I get two convictions,
but I'll see to it that you both get early parole.If you both remain
silent, I'll have to settle for token sentences on firearms possession
charges.If you wish to confess, you must leave a note with the
jailer before my return tomorrow morning.But the outcome obtained when both confess is worse for each
than the outcome they would have obtained had both remained silent.Puzzles with the structure of the prisoner's dilemma were devised and
discussed by Merrill Flood and Melvin Dresher in 1950, as part of the
Rand Corporation's investigations into game theory (which Rand pursued
because of possible applications to global nuclear strategy).Since then the flow has shown no signs of abating.PD With Ordinal Payoffs
2.The Centipede and the Finite IPD
13.Iteration With Error
16.PDs and Social Networks
19.For each possible pair of moves, the
payoffs to Row and Column (in that order) are listed in the
appropriate cell.It is now easy to see that we have the
structure of a dilemma like the one in the story.Then Row gets R for cooperating and T
for defecting, and so is better off defecting.By symmetry
D also strictly dominates C for
Column.In
standard treatments, game theory assumes rationality and common
knowledge.Each player is rational, knows the other is rational, knows
that the other knows he is rational, etc.It is also worth noting that the outcome
(D, D) of both players defecting is
the game's only strong nash equilibrium, i.Then, for each player, although D does not strictly
dominate C, it still weakly dominates in the
sense that each player always does at least as well, and sometimes
better, by playing C.Under these conditions it still
seems rational to play D, which again results in the
payoff that neither player prefers.Asymmetry
Without assuming symmetry, the PD can be represented by using
subscripts r and c for the payoffs to Row and Column.If we assume that the payoffs are ordered as before for each player,
i.D, D) of both players
making this move is worse for each than (C,
C).If these conditions all obtain the argument for dilemma goes through
as before.Defection strictly dominates cooperation for each player,
and (C, C) is strictly preferred by
each to (D, D).C, C) weakly better than
(D, D) (i.Now suppose we drop the first inequality of either a or
b (but not both).Column,
knowing that Row is rational, knows that Row will defect, and so, by
the remaining inequality in b, will defect
himself.By c, the resulting
(D, D) is again worse for both than
(C, C).Defection is no longer dominant, because each
player is better off choosing C than
D when the other chooses N.Nevertheless
(D, D) is still the unique
equilibrium.Multiple Players
Most of those who maintain that the PD illustrates something important
about morality seem to believe that the basic structure of the game is
reflected in situations that larger groups, perhaps entire societies,
face.But
it is unlikely that we face many situations of this structure.More generally, there
is some social benefit B that each member can achieve if
sufficiently many pay a cost C.If enough of her neighbors get the vaccine, each person may
be protected without assuming the risks.First, even if each
player's moves are entirely independent of the others, the
alternatives represented by the columns in the commons matrix above
are no longer independent of the alternatives represented by the
rows.My choosing C necessarily increases the chances
that more than n people will choose C.Provided that n is large, however, it
would seem that this effect could be ignored and we could assume, for
practical purposes, that the payoff matrix is like the previous one.Similarly, whereas we saw in the original PD that mutual defection was
the only nash equilibrium, this game has two equilibria.One is
universal defection, since any player unilaterally departing from that
outcome will move from payoff 0 to C.But a second is the
state of minimally effective cooperation, where the number of
cooperators just exceeds the threshold.This might suggest that the
tragedy of the commons is less tragic than the PD, but in real life
situations, it would seem unlikely that the participants would know
when they are at the equilibrium point of minimally effective
cooperation.But in the commons
game the only pareto optimal outcomes are those of minimally effective
cooperation.In the medical
example it may seem best to vaccinate everyone.Mutual cooperation is identical to minimally effective
cooperation and therefore is both an equilibrium outcome and a pareto
optimal outcome.Now suppose, in addition, that, once the threshold of
effective cooperation has been exceeded, any benefit one gets from
from the presence of an additional cooperator is exceeded by one's
cost of cooperation and that the costs of ineffective cooperation are
genuine, i.We then have a tragedy of the commons game, which
presents a familiar dilemma: defection benefits an individual in every
circumstance (except the one where exactly t others
cooperate) but everybody is better off in any state of effective
cooperation than in any state without it.This account could be
easily be modified to allow threshold of minimally effective
cooperation to differ from one individual to another (i's
clean water requirements might be more stringent than j's for
example) or to allow B to be defined everywhere (thus
eliminating the threshold, so that we always benefit from another's
cooperation).The resulting game would still have its PD flavor.My temptation is to enjoy
some benefits brought about by burdens shouldered by others.My
temptation is to benefit myself by hurting others.If all fill out their applications
honestly, they all have an equal chance of being hired.If one lies,
however, he can ensure that he is hired while, let us say, incurring a
small risk of being exposed later.Thus a lone liar, by reducing the others' chances of
employment from slim to none, raises his own chances from slim to
sure.Nevertheless, the liars seem to be foul dealers rather than free
riders.Their
conditions might, however be a plausible model for certain public
good dilemmas.This outlook has the
advantage of focusing attention on the PD quality of the
game.Defection dominates cooperation, while universal cooperation is
unanimously preferred to universal defection.Michael Taylor goes
even further in this direction.Taylor's main concern is with the
iterated version of this game, a topic that will be addressed in
future editions of this entry.Single Person Interpretations
The PD is usually thought to illustrate conflict between individual
and collective rationality, but the multiple player form (or something
very similar) has also been interpreted as demonstrating problems
within standard conceptions of individual rationality.You are attached to the device
and given the following choice every day for ten years: advance the
device one setting and collect a thousand dollars, or leave it where
it is and get nothing.First, the moves of the
players are sequential rather than simultaneous (and each player has
knowledge of preceding moves).Second, there is the matter of
gradation.It is reasonable to suppose that each acts
in the knowledge of how others have acted before.It is also reasonable to
suppose that addition of one can of garbage to the lake has no
perceptible effect on water quality, and therefore no effect on the
welfare of the residents.It seems appropriate,
however, to separate this issue from that raised in the standard PD.Gradations that are imperceptible individually, but weighty en masse
give rise to intransitive preferences.PD is suggested in Kavka,
1991.Let us imagine
that I am hungry and considering buying a snack.Buy a scoop of orange sherbet.It
is also possible, Kavka suggests, that my inner conflicts are resolved
as if they were a result of strategic interaction among rational
subagents.In this case, Arnold and Eppie can each choose either to
insist on getting their way (I) or to
acquiesce to a compromise (A).Examination of the table and preference orderings confirms that we
again have an intrapersonal PD.Cardinal Payoffs
If the game specifies absolute (as opposed to relative) payoffs, then
universal cooperation may not be a pareto optimal outcome even in the
two person PD.The four outcomes entered in the matrix of the second
section are represented by the labeled dots.Conditions PD3a and PD3b
ensure that (C, D) and
(D, C) lie northwest and southeast
of (D, D), and PD3c is reflected in
the fact that (C, C) lies northeast
of (D, D).Suppose first that
(D, D) and (C,
C) lie on opposite sides of the line between
(C, D) and (D,
C), as in the graph on the left.In the graph on the
left the payoff for universal cooperation (with probability one) is
pareto optimal among the payoffs for all mixed strategies.In the
graph on the right, however, where both (D,
D) and (C, C) lie
southwest of the line between (C, D)
and (D, C), the story is more
complicated.Here the payoffs of the feasible outcome lie within a
figure bounded on the northeast by three distinct curve segments, two
linear and one concave.It is important to note that we are talking about
independent mixed strategies here.If they were able to
correlate their mixed strategies, so as to ensure, say
(C, D) with probability p
and (D, C) with probability
p*, the set of feasible solutions would extend up to (and
include) the dotted line between (C,
D) and (D, C).The
point here is that, even confined to independent strategies, there are
some games satisfying PD3 in which both players can both do better
than they do with universal cooperation.PD in which universal
cooperation is pareto optimal may be called a pure PD.This
phenomenon is identified in Kuhn and Moresi and applied to moral
philosophy in Kuhn 1996.Rapoport, Chammah and Axelrod who employed
it).Since the reward
payoff exceeds the punishment payoff, I should cooperate.More
generally, even if my accomplice is not a perfect replica, the odds of
his cooperating are greater if I cooperate and the odds of his
defecting are greater if I defect.These arguments closely resemble the
arguments for two positions on the Newcomb Problem, a puzzle
popularized among philosophers in Nozick.The extent of the
resemblance is made apparent in Lewis.The Newcomb Problem asks us to
consider two boxes, one transparent and one opaque.In the transparent
box we can see a thousand dollars.We know before
choosing that a reliable predictor of our behavior has put a million
dollars in the opaque box if he predicted we would take the first
choice and left it empty if he predicted we would take the second.To
see that each player in a PD faces a Newcomb problem, consider the
following payoff matrix.PD (and any such PD can be represented
in this form).Two
boxing is a dominant strategy: two boxes are better than one
whether the first one is full or empty.See Hurley, however,
for an argument that the two puzzles are significantly different.In a
PD (of either the ordinary or the Newcombized variety) each player
knows that the other is rational and that the other ranks the outcomes
in the ways described.This, Hurley argues, opens a possibility for
cooperative joint action that is absent in the original Newcomb
problem.This
apparent conflict has led some to suggest that standard decision
theory needs to be refined in cases in which an agent's actions
provide evidence for, without causing, the context
in which he is acting.C1)
is the conditional probability that player Two cooperates given that
Player One cooperates).C1)
will be close to zero.One were to cooperate, Two would
also cooperate.Lewis argues that the link to
the PD suggests that situations where the two decisions diverge are
not so unusual, and recent writings on causal decision theory contain
many examples far less bizarre than Newcomb's problem.One reason for the change in nomenclature is to
distinguish these ideas from an experimental literature reporting on
PD games played with real (identical or fraternal) twins.See, for
example, Segal and Hershberger.It turns out that twins are
more likely to cooperate in a PD than strangers, but there seems to be
no suggestion that the reasoning that leads them to do so follows the
controversial arguments presented above.This game is known as the Stag Hunt.It might provide a
better model for situations where cooperation is difficult, but still
possible, and it may also be a better fit for other roles sometimes
assigned to the PD.The fable dramatizing the game and providing its name, gleaned from a
passage in Rousseau's Discourse on Inequality, concerns a
hunting expedition rather than a jail cell interrogation.Two hunters
are are looking to bag a stag.Success is uncertain and, if it comes,
require the efforts of both.On the other hand, either hunter can
forsake his partner and catch a hare with a good chance of success.Stag Hunt is no longer much of a
temptation, but we retain the payoff terminology for ease of
exposition.Either way, the essence of the Stag Hunt remains.There
are two equilibria, one unanimously preferred to the other.It
is clear that if I am certain that my partner will hunt stag I should
join him and that if I am certain that he will hunt hare I should hunt
hare as well.The matrix above provides one example.Since the sucker payoff is the worst payoff in a Stag Hunt,
this principle suggests that any Stag Hunt presents a
dilemma.Stag Hunt no mixed
strategies are ever preferred to mutual cooperation.This might be a good model for cooperative activity in which
success requires full cooperation.Stag Hunt
Dilemmas in an extreme form.Everyone would benefit if all cooperate,
but only a very trusting fool would think it rational to cooperate
himself.The cooperative outcome in the Stag Hunt can be assured by many of the
same means as are discussed here for the PD.As might be expected,
cooperation is somewhat easier to come by in the two person Stag Hunt
than in the two person PD.Here we
eliminate the requirement that the two players move simultaneously.Consider the situation of a firm whose sole competitor has just
lowered prices.We
can think of these as situations in which one player has to choose to
cooperate or defect after the other player has already made a similar
choice.Careful discussion of an asynchronous PD example, as Skyrms (1998) and
Vanderschraaf recently note, occurs in the writings of David Hume,
well before Flood and Dresher's formulation of the ordinary PD.Figure 2
Here, time flows to the right.The moves and the payoffs to each player are exactly as in the
ordinary PD, but here Player Two can choose his move according to what
Player One does.Player One knows that if he were to choose
C on the first move, Player Two would choose
D on the second move (since she prefers the
temptation to the reward), so he would himself end up with the sucker
payoff.If Player One were to choose D, Player Two
would still choose D (since she prefers the
punishment to the sucker payoff), and he would end up with the
punishment payoff.Since he prefers the punishment payoff to the
sucker payoff, Player One will choose D on the first
move and both players will end up with the punishment payoff.The result is a two player
game with the following matrix.The game is not, however, a dominance PD.To preserve the symmetry between the players that characterizes the
ordinary PD, we may wish to modify the asynchronous game.First each player chooses a first
move (C or D) and a second move (
Cu, Du, I or
O).Next a referee determines who moves first, giving
each player an equal chance.Finally the outcome is computed in the
appropriate way.It is straightforward, but tedious, to
calculate the entire eight by eight payoff matrix.The
sole nash equilibrium occurs when both players adopt the strategy
(D, Du), thereby achieving the
inferior payoffs of (P, P).It may be worth noting that an asynchronous version of the Stag Hunt,
unlike the PD, presents few issues of interest.If he hunts hare on day one, she should do
likewise on day two.Thus there may be some
theoretical interest in investigations of PDs with transparent
players.In
RG, Column has the same moves as in game G
and Row can choose any function that assigns C or
D to each of Column's possible moves.Similarly in
CG, Row has the same moves as in G and
Column has a new set of conditional moves.Notice
that this last strategy is tantamount to Danielson's reciprocal
cooperation.The lesson of all this for rational action is not clear.How do they decide what
level game to play?But
why should either player expect the intention to be carried out if
there is benefit in ignoring it?Conditional strategies have a more convincing application when we take
our inquiry as directed, not towards playing the PD, but as designing
agents who would play it well with a variety of likely opponents.This
is the viewpoint of Danielson.Howard for an earlier
enlightening discussion of this viewpoint.Danielson does not limit himself a priori to strategies
within Howard's hierarchy.Programs implementing I and
O in a straightforward way are not likely to succeed
because when paired with each other they will be incoherent.In Howard's scheme we could
compare a conditional strategy with all the possible alternatives of
that level.Here, where any two programs can be paired, that approach
is senseless.Nevertheless, certain programs seem to do well when
paired with a wide variety of players.One is a version of the
strategy that Gauthier has advocated as constrained
maximization.It is not
clear how a program implementing it would move (if indeed it does
move) when paired with itself.Danielson is able to construct an
approximation to constrained maximization, however, that does
cooperate with itself.It cooperates with Cu and itself and it
defects against Du.If it is coherently paired it
seems guaranteed a payoff no worse than P.Again, it is not clear that the strategy (as
formulated above) allows it to cooperate (or make any move) with
itself, but Danielson is able to construct an approximation that does.The (approximate) reciprocal cooperation does as well as
(approximate) constrained maximization against itself,
Du and constrained maximization.Against
Cu it does even better, getting T where
constrained maximization got only R.Finite Iteration
Many of the situations that are alleged to have the structure of the
PD, like defense appropriations of military rivals or price setting
for duopolistic firms are better modeled by an iterated version of the
game in which players play the PD repeatedly, retaining access at each
round to the results of all previous rounds.An IPD can be represented in extensive form by a tree diagram like
the one for the farmer's dilemma above.Figure 3
Here we have an IPD of length two.The payoffs to each
of the two players (obtained by adding their payoffs for the two
rounds) are listed at the end of each path through the tree.In a game like this, the notion of nash equilibrium loses some of its
privileged status.Recall that a pair of moves is a nash equilibrium
if each is a best reply to the other.Let us extend the notation used
in the discussion of the asynchronous PD and let Du
be the strategy that calls for defection at every node of an IPD.It
is easy to see that Du and Du form a
nash equilibrium.But against Du, a strategy that
calls for defection unless the other player cooperated at, say, the
fifteenth node, would determine the same play (and therefore the same
payoffs) as Du itself does.The components that call
for cooperation never come into play, because the other player does
not cooperate on the fifteenth (or any other) move.Similarly, a
strategy calling for cooperation after the second cooperation by
itself does equally well.If Player One had cooperated in the past, that would
still provide no good reason for him to cooperate now.In games of the first kind, one can prove by an argument known
as backward induction that Du,
Du is the only subgame perfect equilibrium.Suppose
the players know the game will last exactly n rounds.PD, and they will
defect.By repeating this argument sufficiently
many times, the rational players deduce that they should defect at
every node on the tree.For a small
sample, see Bovens, Kreps et al, Kreps and Wilson, Pettit and
Sugden, Sobel 1993 and Binmore 1997).Thus the argument for continual
defection in the IPD of fixed length depends on complex iterated
claims of certain knowledge of rationality.C, C), Player One will
choose D despite never having done so before.For (with plausible assumptions) one way to ensure that a rational
player will doubt one's own rationality is to behave irrationally.Player Two would rationally react so that they can achieve
mutual cooperation in almost all rounds.So our assumptions seem to
imply both that Player One should continually defect and that she
would do better if she didn't.IPD can be raised in
even starker form by a somewhat simpler game.Consider a PD in which
they punishment payoff is zero.Now iterate the asynchronous version
of this game a fixed number times.The game ends when the stack runs out or one of the
players takes two bills (whichever comes first).Infinite Iteration
One way to avoid the dubious conclusion of the backward induction
argument without delving too deeply into conditions of knowledge and
rationality is to consider infinitely repeated PDs.No human agents
can actually play an infinitely repeated game, of course, but the
infinite IPD has been considered an appropriate way to model a series
of interactions in which the participants never have reason to think
the current interaction is their last.In this setting a pair of
strategies determines an infinite path through of the game tree.See Binmore 1992, page 365 for further
justification.Mathematically, it makes little
difference whether p is regarded as a probability of
continuation or a discount on payoffs.The value of cooperation at a
given stage in an IPD clearly depends on the odds of meeting one's
opponent in later rounds.PD, and the
value of defection increases.There is an observation, apparently originating in Kavka, 1983, and
given more mathematical form in Carroll, that the backward induction
argument applies as long as an upper bound to the length of the game
is common knowledge.For example, since
shopkeeper Jones cannot make more than one sale a second and since he
will live less than a thousand years, he and customer Smith can
calculate (conservatively) that they cannot possibly conduct more than
1012 transactions.Note first that, in an indefinite IPD as described
above, there can be no upper bound on the length of the game.If the interaction of Smith and Jones
were modeled as an indefinite IPD, therefore, the probability of their
interacting in a thousand years would not be zero, but rather some
number greater than pk where p
is the probability of their interacting again now and k is
the number of seconds in a thousand years.As long as p always remains
greater than zero, however, it remains true that there can be no upper
bound on the number of possible interactions, i.Suppose, on the
other hand, that there was a number n such that that
there was zero probability of the game's continuing to stage
n.Given the standard common knowledge assumptions that we have been
making, the players would know this value of i, and the IPD
would be one of fixed length, and not an indefinite IPD at all.In the
case of the shopkeeper and his customer, we are to suppose that both
know today that their last interaction will occur, let's say, at noon
on June 10th, 2020.If the players know all the values of
pi from the outset, then, as long as the
value of pi becomes and remains
sufficiently small, they (and we) can compute a stage k at
which the risk of future punishment and the chance of future reward no
longer outweighs the benefit of immediate defection.So they know
their opponent will defect at stage k, and the induction
begins.Carroll argument, however, only
further exposes the implausibility of its assumptions.Furthermore each is expected to
believe that the other has made this computation, and that the other
expects him to have made it, and so on.The iterated version of the PD was discussed from the time the game
was devised, but interest accelerated after influential publications
of Robert Axelrod in the early eighties.More significant than TFT's
initial victory, perhaps is the fact that it won Axelrod's second
tournament, whose sixty three entrants were all given the results of
the first tournament.As a further
demonstration of the strength of TFT, he calculated
the scores each strategy would have received in tournaments in which
one of the representative strategies was five times as common as in
the original tournament.TFT received the highest
score in all but one of these hypothetical tournaments.An unforgiving rule is
incapable of ever getting the reward payoff after its opponent has
defected once.Suggestive as Axelrod's discussion is, it is worth noting that the
ideas are not formulated precisely enough to permit a rigorous
demonstration of the supremacy of TFT.One doesn't
know, for example, the extent of the class of strategies that might
have the four properties outlined, or what success criteria might be
implied by having them.Since
TFT is itself one such strategy this implies that
TFT forms a nash equilibrium with itself in the space
of all strategies.TFT is, in general, not subgame
perfect.Iteration With Error
In a survey of the field several years after the publication of the
results reported above, Axelrod and Dion, chronicle several successes
of TFT and modifications of it.One
such case, noted in the Axelrod and Dion survey, is when attempts are
made to incorporate the plausible assumption that players are subject
to errors of execution and perception.There are a number of ways this
can be done.Molander 1985 demonstrates that strategies that mix
TFT with Cu do approach a payoff of
R as the probability of error approaches zero.Following Nowak and Sigmund, we label this strategy
generous TFT, or GTFT.The idea that the presence of imperfection induces greater forgiveness
or generosity is only plausible for low levels of imperfection.Kollock seems to confirm that at high levels of imperfection, more
stinginess is better policy than more forgiveness.But Bendor, Kramer
and Swistak note that the strategies employed in the Kollock
simulation are not representative and so the results must be
interpreted with caution.Tats (which cooperates unless defected
against twice in a row).It was also beaten, however, by two versions
of Downing, a program that bases each new move on its
best estimate how responsive its opponent has been to its previous
moves.Strictly
speaking, Pn is not fully
specified until an initial probability of cooperation is given, but
for most purposes the value of that parameter becomes insignificant in
sufficiently long games and can be safely ignored.Each move for the latter depends on only on its
opponent's last move, whereas each move for
Pn is a function of the entire
history of previous moves of both players.Against responsive strategies, like other Pavlovian
strategies and TFT,
Pn and its opponent eventually
reach a state of (almost) constant cooperation.It should be noted, however, that
when (deterministic) TFT plays itself, no training
time at all is required, whereas when a Pavlovian strategy plays
TFT or another Pavlov, the training time can be
large.One advantage of the evolutionary
versions of the IPD discussed in the next section is that they permit
more careful formulation and evaluation of success criteria.Evolution
Perhaps the most active area of research on the PD concerns
evolutionary versions of the game.Thus success in an evolutionary PD (henceforth
EPD), requires doing well with other successful strategies, rather
than doing well with a wide range of strategies.It is assumed that the
size of the entire population stays fixed, so that births of more
successful strategies are exactly offset by deaths of less successful
ones.V),
where Vi is the score of
si in the previous round and V
is the average of all scores in the population.Thus every strategy
that scores above the population average will increase in number and
every one that scores below the average will decrease.Other rules of evolution are possible.Since rational players would presumably
switch only to strategies that received the highest payoff in previous
rounds, only the highest scoring strategies would increase in
numbers.Discussion here, however, will
primarily concern EPDs with the proportional fitness rule.Changing this third feature might well be expected to hurt
TFT.For a large growth in the TFT
population would make it possible for mutants employing more naive
strategies like Cu to regain a foothold, and the
presence of these naifs in the population might favor nastier
strategies like Du over TFT.Cu, Du,
TFT, and Cp are
R(1,1,1), R(0,0,0),
R(1,1,0), and R(p,
p, p).To capture the inevitability of error, Nowak
and Sigmund exclude the deterministic strategies, where p and
q are exactly 1 or 0, from their tournaments.As before, if
the game is sufficiently long (and p and q are not
integers), the first move can be ignored and a reactive strategy can
be identified with its p and q values.Those strategies
R(p, q) closest to
R(0,0) thrived while the others perished.But an embattled minority remains and fights
back.Slowly at first, but gathering
momentum, the reciprocators come back, and the exploiters now
wane.On the basis of their tournaments among reactive strategies, Nowak and
Sigmund conjectured that, while TFT is essential for
the emergence of cooperation, the strategy that actually underlies
persistent patterns of cooperation in the biological world is more
likely to be GTFT.The strategies
considered in the second series allowed each player to base its
probability of cooperation on its own previous move as well as its
opponent's.Again, we can ignore the probability of
defecting on the first move as long as the
pis are not zero or one.The
results are quite different than before.Kraines and Kraines
had been somewhat dismissive of P1.One reason may be that
P1 helps to make its environment
unsuitable for its enemies.Thus, although TFT fares less
badly against Du than P1
does, P1 is better at keeping its
environment free of Du.Simulations in a universe of deterministic strategies yield results
quite different than those of Nowak and Sigmund.For example,
P1 is represented by the machine pictured
below.It begins in the
leftmost state.The arrow leading from the
left to the right circle indicates that machine defects (enters the
D) after it has cooperated (been in the
C state) and its opponent has defected (the arrow is
labeled by d).It turns out that these are exactly the
deterministic versions of the S strategies of Nowak
and Sigmund.Since the strategies are deterministic, we must
distinguish between the versions that cooperate on the first round and
those that defect on the first round.Each of the other
S(p1, p2,
p3, p4) where
p1, p2,
p3, p4 are either zero or one
represent unique strategies.This is a strategy whose
imperfect variants seem to have been remarkably uncompetitive for
Nowak and Sigmund.It has been frequently discussed in the game theory
literature under the label GRIM or
TRIGGER.It cooperates until its opponent has
defected once, and then defects for the rest of the game.The explanation for the discrepancy between
GRIM's performance for Linster and for Nowak and
Sigmund probably has to do with its sharp deterioration in the
presence of error.Selten 1983, includes an example of a game
with no evolutionarily stable strategy, and Selten's argument that
there is no such strategy clearly applies to the EPD
and other evolutionary games.Unsurprisingly,
the paradox is resolved by observing that the three groups of authors
each employ slightly different conceptions of evolutionary
stability.Readers who wish to compare these with some others
that appear in the literature may consult the following brief
guide:
Concepts of Stability in Evolutionary Games.Here, and in what follows, the notation
V(i, j) indicates the
payoff to strategy i when i plays
j.MS says that any invaders do strictly worse
against the natives than the natives themselves do against the natives
or else they get exactly the same payoff against the natives as the
natives themselves do, but the native does better against the invader
than the invader himself does.For any strategy i in the IPD (or indeed in any
iterated finite game), however, there are strategies
j different from i such that
j mimics the way i plays when it
plays against i or j.There may be good reason to restrict the available
strategies.For example, if the players are assumed to have no
knowledge of previous interactions, then it may be appropriate to
restrict available strategies to the unconditional ones.Indeed, this is the kind of evolutionary game that Maynard
Smith himself considered.This does not particularly vindicate any of
the strategies discussed above, however.One
way to distinguish among the strategies that meet BS is by the size of
the invasion required to overturn the natives, or, equivalently, by
the proportion of natives required to maintain stability.They maintain that this result does
allow them to begin to provide a theoretical justification for
Axelrod's claims.Bendor and Swistak's results must be interpreted with some care.If the
number of generations is large compared with the original population
(as it often is in biological applications), a population that is
initially composed entirely of players employing the same maximally
robust strategy, could well admit a sequence of small invading groups
that eventually reduces the original strategy to less than half of the
population.At that point the original strategy could be
overthrown.Since the simulations
required imperfection and since they generated a sequence of mutants
vastly larger than the original population, there is no real
contradiction here.Nevertheless the discrepancy suggests that we do
not yet have a theoretical understanding of EPDs sufficient to predict
the strategies that will emerge under various plausible conditions.If agents are not paired at random, but rather are more likely
to play others employing similar strategies, then cooperative behavior
is more likely to emerge.This may be an array with a
rectangular boundary, for example, or a circle, or surface of a sphere
or surface of a torus with no boundary.SPD than they would be in an ordinary evolutionary
game.As usual, the impetus for looking at spatial SPDs seems to come from
Axelrod.Axelrod's payoffs of 5,3,1 and 0 for T,
R, P and S, do meet this condition.For a narrow range of intermediate
values, we get more of the complicated patterns noted above.Mukherji and Rajan, although
cooperators seem to require lower relative temptation values to thrive
under these conditions, and the level of error must be sufficiently
low.Grim, Mar and St Denis report a number of SPD simulations with a
greater variety of initial strategies.Again, other outcomes are possible.Simulations beginning with a random selection
of a few (viz.TFT with considerably more generosity than
GTFT.Nevertheless SPD models of the
evolution of cooperation in particular geometrical arrangements have
given us some suggestive and pretty pictures to contemplate.PDs and Social Networks
One way to make the idea of local interaction more realistic for some
applications is to let the agents choose the partners with
whom to interact, based on payoffs in past interactions.Initially, as usual, each agent chooses a partner at
random from the remaining members of the population.Since
the cooperators are chosen by both cooperators and defectors, they
play more often than the defectors who play only when they are
chosen.Since they rapidly cease being chosen by cooperators,
however, their returns from interactions with cooperators will be less
than returns from defectors and they will soon limit their choices to
other defectors.It is important to understand here that the learning
algorithm that determines the probability I will interact with agent
a depends on total returns from interacting
with a (or total recent returns from
interacting with a) rather than average
returns from interacting with a.So in the
attenuated game we end up with perfect association: defectors play
defectors and cooperators play cooperators.The social network games considered above are not really evolutionary
PDs in the sense described above.The patterns of interaction evolve,
but the strategy profile of the population remains fixed.Whether cooperation
or defection (or neither) comes to dominate the population under such
conditions depends on a multitude of factors: the values of the
payoffs, the initial distribution of strategies, the relative speed of
the adjustments in strategy and interaction probabilities, and other
properties of those two evolutionary dynamics.See Sober and Wilson or Wilson and Sober for
a history and impassioned defense of this resuscitation.Consider,
for example, a simple version of the haystack model
originally described by John Maynard Smith.Pairs of players from a
large population pair randomly.Each pair colonizes a single
haystack.The pair plays a prisoners dilemma and the payoffs to an
individual determine the number of offspring of that individual in the
next generation.The members
of the colony pair randomly with other members and play the PD for
some fixed number of generations.Then the haystacks are torn down,
the population mixes and random pairs colonize next season's
haystacks.Then, if
Player One cooperates and player Two defects, the payoff to Player One
will be 0, because a cooperator gets 0 offspring in the second, and
any subsequent, generation.The full payoff matrix for the
four generation haystack PD with payoffs 3,2,1, and 0 is given by the
matrix below.Gasparski, Wojciech et al (eds), Social
Agency, New Brunswick, N.Heath and Company (1992).Binmore, Kenneth, Playing Fair: Game Theory and the Social
Contract 1, Cambridge, MA: MIT Press (1994).Binmore, Kenneth, Natural Justice New York, NY: Oxford
Univsity Press (2005).Cambell, Richmond and Lanning Snowden, Paradoxes of
Rationality and Cooperation, Vancouver: University of British
Columbia Press (1985).Danielson, Peter, Artificial Morality: Virtual Robots for
Virtual Games, London: Routledge (1992).Is the Symmetry Argument Valid?Joyce, James, The Foundations of Causal Decision Theory,
Cambridge University Press, (1999).Resher (ed) Essays in Honor of Carl
G.Poundstone, William, Prisoner's Dilemma New York:
Doubleday (1992).Schelling, Thomas, Micromotives and Macrobehavior New
York: Norton (1978).Skyrms, Brian, The Dynamics of Rational Deliberation,
Cambridge, MA: Harvard University Press (1990).Skyrms, Brian, Evolution of the Social Contract,
Cambridge, Cambridge University Press (1996).Coleman
and Morris (eds.Rational Commitment and Social Justice: Essays
for Gregory Kavka, New York, Cambridge University Press
(1998).Backward Induction Without Tears?Sober, Elliot and David Sloan Wilson, Unto Others: The
Evolution and Psychology of Unselfish Behavior, Cambridge, MA:
1998.Second edition published in 2004 by
Palgrave MacMillan, Basingstoke, UK.Taylor, Michael, The Possibility of Cooperation,
Cambridge: Cambridge University Press (1987).Laboratoire d'Informatique Fondamentale de
Lille).Miscellaneous PD Resources
(compiled by Constitution
Society).Recent developments in game theory, especially the award of the Nobel Memorial
Prize in 1994 to three game theorists and the death of A.The publication of The Theory of Games and Economic
Behavior was a particularly important step, of course.But in some ways, Tucker's
invention of the Prisoners' Dilemma example was even more important.In 1950, while
addressing an audience of psychologists at Stanford University, where he was a visiting
professor, Mr.Tucker's simple explanation
has since given rise to a vast body of literature in subjects as diverse as philosophy,
ethics, biology, sociology, political science, economics, and, of course, game theory.If each confesses and implicates the other, both will go to prison
for 10 years.However, if one burglar confesses and implicates the other, and the
other burglar does not confess, the one who has collaborated with the police will
go free, while the other burglar will go to prison for 20 years on the maximum charge.The table is read like this: Each prisoner chooses one of the two strategies.Then I get 20 years if I don't confess, 10 years if I do, so in that case it's best
to confess.On the other hand, if Bob doesn't confess, and I don't either, I get
a year; but in that case, if I confess I can go free.Either way, it's best if I
confess.DEFINITION Dominant Strategy: Let an individual player in a game
evaluate separately each of the strategy combinations he may face, and, for each
combination, choose from his own strategies the one that gives the best payoff.For there are many interactions in
the modern world that seem very much like that, from arms races through road congestion
and pollution to the depletion of fisheries and the overexploitation of some subsurface
water resources.These are all quite different interactions in detail, but are interactions
in which (we suppose) individually rational action leads to inferior results for
each person, and the Prisoners' Dilemma suggests something of what is going on in
each of them.In the Prisoners' Dilemma, the two prisoners interact only once.Repetition of
the interactions might lead to quite different results.Compelling as the reasoning that leads to the dominant strategy equilibrium may
be, it is not the only way this problem might be reasoned out.Perhaps it is not
really the most rational answer after all.Prisoners' Dilemma is a game which has been and continues to be studied by people in a variety of disciplines, ranging from biology through sociology and public policy.Prisoners' Dilemma, including both original studies of strategies and discussion of the game's broader significance is Robert Axelrod's The Evolution of Cooperation (Basic Books, NY, 1984).Still more recent is "The Arithmetics of Mutual Help", by Martin Nowak, Robert M.EACH, The "Evolution of Altruistic and Cooperative Behaviors" is a modelling and curriculum project, using StarLogo.Tit for tat in heterogeneous populations.Hours() * 3600 + newCurrentTime.Minutes() * 60 + currentTime.Are You an Author or Publisher?Copyright 1993 Reed Business Information, Inc.From Library Journal
This very readable book is partly a biography of John von Neumann, partly a nontechnical history of the branch of mathematics known as game theory, and partly a description of some of the paradoxical findings that arise from that theory.Those sections that deal with game theory use no mathematics beyond simple arithmetic and are thus fascinating, thought provoking, and easily accessible to the layperson.For all biography and science collections.This text refers to an out of print or unavailable edition of this title.Books (See Bestsellers in Books)Popular in this category: (What's this?Would you like to update product info or give feedback on images?The Computer and the Brain: Second Edition (Mrs.The Paradox of God and the Science of Omniscience
by Clifford A.CyberPhilosophy: The Intersection of Philosophy and Computing (Metaphilosophy)
by James H.The Transformation of the U.What Do Customers Ultimately Buy After Viewing Items Like This?Click on a tag to find related items, discussions, and people.Why not be the first to suggest a search for which it should appear?See all 350 customer reviews...The easiest way to shoot video reviews.William Poundstone is in his element when he's writing about stuff like this.In part it's a biography (intellectual and otherwise) of John von Neumann, one of the greatest mathematicians of the twentieth century.You pretty much have to be William Poundstone to weave all this together into a coherent and readable narrative.There's something here for everybody.If even one of the strands in this tale sounds engaging to you, you can rest assured that Poundstone will manage to keep you engaged in the other two as well.Look for his other books too."Thanks for the valuable feedback you provided to other Amazon.Your vote will be counted and will appear on the product page within 24 hours."This book spends, as much time on history and biography as it does on what Game Theory is about, so this work would seem to be most appropriate to those who are new to the material.The historical and biographic aspects of the book were not new, so there were of less interest to me.The various games that are explained and, "played", for the reader actually utilize little in the way of math.Game Theory in practice is about the number of participants, the choices they have, how the games should rationally be played, and how there are played when people replace theory.The results of these games are applicable to daily life, whether it explains how a network will decide the placement of their commercials, why a person will stand in a line of unknown length, or pay more than the true value of an item (like a dollar bill).Peoples behavior often crosses from the irrational to the absurd, and many of these games will point out courses of action almost all readers will have taken at one time or another, when the rational decision was the opposite of what they chose to do.The book is also a good primer for further reading on Bertrand Russell, John Nash the subject of the movie, "A Beautiful Mind", and John von Neumann, who many considered the most brilliant man alive during his career, and many other great scientists of the 20th Century.There is also review of the development of both the atomic and hydrogen bombs, and the very surprising groups of people that either supported their development and use, and those that were diametrically opposed.There is also some discussion on how Game Theory was and is used to make decisions on a global scale, and also where Game Theory falls short of some of its initial promise.Was this review helpful to you?Neither are convered in extreme detail, but that isn't bad.In fact, it makes this book an enjoyable casual read.The Prisoner's Dilemma is one of those ideas that's so simple and so profound that it deserves to be studied by everyone who's interested in human nature.Not his best
I liked Poundstone's Labyrinths of Reason: Paradox, Puzzles, and the Frailty of Knowledge much more than this book.Layman's intro to Game Theory
I enjoyed this book and learned a lot.Well written for the everyday reader
As someone familiar with the concept of game theory especially the prisoner's dilemma, I found this book to be particularly informative, on a layman's level.And so, our understanding of physics is enlightened by the interplay...Economic books for everyone: A list by James D.Lose the Smoke, Keep the Fire
Who will you be without that cigarette?If you need help or have a question for Customer Service, contact us.Would you like to update product info or give feedback on images?Track your recent orders.Bibliography on the Evolution of Cooperation.The Complexity of Cooperation.The Evolution of Cooperation.Less Cooperation in Iterated Prisoner's Dilemma.Strategies for Iterated Prisoners Dilemma.Games and Economic Behavior.Systems, Man and Cybernetics, IEEE Press, Piscataway, NJ, USA, pp."Prisoner's Dilemma" (Axelrod, 1984).The two players in the game can choose between two moves, either "cooperate" or "defect".If both defect, both lose (or gain very little) but not as much as the "cheated" cooperator whose cooperation is not returned.However, the police does not have sufficient proof in order to have them convicted.However, if one of them betrays the other one, by confessing to the police, the defector will gain more, since he is freed; the one who remained silent, on the other hand, will receive the full punishment, since he did not help the police, and there is sufficient proof.If both betray, both will be punished, but less severely than if they had refused to talk.Even if an altruistic wolf would kill a rabbit and give it to another wolf, and the other wolf would do nothing in return, the selfish wolf would still have less to eat than if he had helped his companion to kill a deer.Long term cooperations can only evolve after short term ones have been selected: evolution is cumulative, adding small improvements upon small improvements, but without blindly making major jumps.The problem is that if both actors are rational, both will decide to defect, and none of them will gain anything.See also:
an interactive implementation of the Prisoner's dilemma where you can play the game yourself
Bjoern Brembs' review on the iterated Prisoner's Dilemma:
Heylighen F.It helps us understand what governs the balance between cooperation and competition in business, in politics, and in social settings.In the traditional version of the game, the police have arrested two suspects and are interrogating them in separate rooms.Each can either confess, thereby implicating the other, or keep silent.Thus, confession is the dominant strategy (see Game Theory) for each.The concept of the prisoners' dilemma was developed by Rand Corporation scientists Merrill Flood and Melvin Dresher and was formalized by a Princeton mathematician, Albert W.The prisoners' dilemma has applications to economics and business.Cola and Pepsi, selling similar products.Each must decide on a pricing strategy.Collusion to keep prices high, for example, is not in society's interest because the cost to consumers from collusion is generally more than the increased profit of the firms.Similarly cooperation among prisoners under interrogation makes convictions more difficult for the police to obtain.The cheater's reward comes at once, while the loss from punishment lies in the future.If players heavily discount future payoffs, then the loss may be insufficient to deter cheating.Thus, cooperation is harder to sustain among very impatient players (governments, for example).Punishment won't work unless cheating can be detected and punished.Punishment is usually easier to arrange in smaller and closed groups.Thus, industries with few firms and less threat of new entry are more likely to be collusive.Punishment can be made automatic by following strategies like "tit for tat," which was popularized by University of Michigan political scientist Robert Axelrod.Here, you cheat if and only if your rival cheated in the previous round.Both or all players know that cheating is the dominant strategy in the last play.But in practice we see some cooperation in the early rounds of a fixed set of repetitions.The reason may be either that players don't know the number of rounds for sure, or that they can exploit the possibility of "irrational niceness" to their mutual advantage.Avinash Dixit is the John J.Sherred Professor of Economics at Princeton University.Barry Nalebuff is the Milton Steinbach Professor of Management at Yale University's School of Organization and Management.Thinking Strategically: A Competitive Edge in Business, Politics, and Everyday Life.Photo courtesy of author.Conscience is the inner voice that warns us somebody may be looking.An interesting endorsement for Obama today.You know, there is more to blogging than being unprofessional.If you have looked at other bloggers, you would notice the use of hyperlinks."Pelosi's BFF George Miller," you could use a hyperlink to point to an article describing their relationship.Access to links to the sources of stated assertions is one of the reasons that blogs are so popular.You should write a piece about that.High School Girl mentality that seems to be so popular among the Inside Washington pudit class.If so, on the Ashamed to Proud spectrum, how do you feel about it.How do you think your High School Journalism teachers feel about it?Senator Dodd is going to Filibuster this.If you would contact your Senators and ask them to fulfill their oaths to uphold the Constitution, that would be nice.The link above has you fill out a form; IT figures out who your senators are and forwards the message to them.There are a lot of good reasons to be against the bill.Chris Dodd is going to Filibuster this on Monday.If you want to give Senator Dodd an encouraging word, you can do it here:firedoglake.Borgarello says, and pollutants that come in contact with the surface of the cement are oxidized.Hazardous nitrogen oxides and sulfur oxides, for example, are transformed into harmless nitrates or sulfates, which simply rinse off the building with rainwater. |
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![Peter Gabriel : [1989] Passion Music for The Last Temptation Of Christ](/covers/37/3780/alb_85006_th.jpg "Peter Gabriel : [1989] Passion Music for The Last Temptation Of Christ")
