Strategies, Rationality, and Game Theory in the

Philosophy of War


Randall R. Dipert

C.S. Peirce Professor of American Philosophy

Department of Philosophy

SUNY at Buffalo




In this paper I would like to develop three observations about current work in the philosophy of war:


1.  The contemporary philosophy of war shows little awareness of, and has not incorporated, major contemporary results of game theory.  These results have have had a major influence in political science and economics.   Relevant research in game theory includes most notably game-theoretic  studies of Robert Axelrod but also Nobel prize-winning work by John Nash (Nobel, 1994) and the work of Robert J. Aumann and Thomas C. Schelling (Nobel, 2005)[1].   This oversight is somewhat surprising since recent discussions in evolutionary biology, some results of which are themselves products of game theory (by Richard Dawkins, John Maynard Smith, and others), have been discussed in moral theory—notably a recent discussion of the possible evolutionary naturalness of altruism and Humean sympathy.


2.  Contemporary philosophy of war has shown little patience with what is ordinarily called Geopolitical Realism (Realpolitik), either ignoring it, rejecting it out of hand, or defining it only in a philosophically unpalatable version of Military Realism: that there is no morality whatsoever in war.  Probably the majority of international affairs experts are geopolitical realists to some degree.


3.  Contemporary philosophy of war has scarcely discussed in a thorough way the morality of preventive war, (a) often distorting historical attitudes to preventive war (e.g., saying that traditional philosophy of war has uniformly rejected it), (b) confusing it with a procedural-legal consideration, namely the legality of preventive war, that is not directly relevant to the morality of war and in any case far more controversial than  usually portrayed and finally, (c) much of the extensive contemporary hostility to war apparently comes from the undemonstrated if initially plausible assumption that a strategy of preventive war, if adopted by all nations, would lead to more wars and destruction than would a ban on preventive war.


     Game theory (since 1948)  has been thought to be relevant to a discussion of war in political science because it appears to capture salient elements of what could be called "idealized conflict" between two or more parties.  As a branch of mathematics, game theory's results have the status of being a priori truths.  The fact that some results have been obtained only through large-scale, randomized computer simulations might give the impression of their being a posteriori and experimental results.  However, some complex a priori results can presently be known by us only through a posteriori observations, and there is nothing inherently mysterious or troubling about this.  To ignore the results of game theory in philosophy because of their superficially experimental and idealized character would be like ignoring the results of arithmetic such as adding and subtraction.  Imagine a discussion of utilitarianism  or another consequentialist theory that began by doubting arithmetic because sometimes, when you "add" two rabbits together, you get—in short order—many more rabbits through reproduction.  Game theory surely has the epistemological character of arithmetic.

     Geopolitical realism often might appear to be amoral or even immoral because of realists' reluctance to place moral considerations as a primary or important consideration.  Indeed, the impatience of some realists with any discussion of moral considerations, such as the human rights of other countries citizens', might give the impression that they believe that moral theory has no place in determining correct strategic policy.  One key tenet of geopolitical realists seems to be roughly that war is a "given" in human existence, and that we must then deal with this fact in a rational way.  There are three versions of this tenet, which might be called the psychological-anthropological thesis about war, the historical argument, and a more rarely articulated rational argument.  If we believe in scientific induction, and notice the frequent appearance of violent conflict whenever humans have organized themselves into groups in human history and shared geographical proximity, the pull of the historical tenet of realism should be very strong.  Both the psychological and the historical versions of the Realist thesis of the ubiquity of war typically have given rise to a sterile philosophical discussion about the "nature" of human beings, and human societies, what such a nature is, and of fatalist or determinist issues of whether we can or should prevent the manifestation of this nature.  Namely, even if human beings and especially organized groups of them are inclined or prone to violent attack on another (whether from greed or from suspicion), surely these forces do not have the forces of a necessary compulsion.  However, the fact that any given individual can overcome this inclination does not detract from the fact that each of us is all but certain to live in a world where other nations and other individuals do initiate violence.   Given this fact, some version of Geopolitical Realism is the most rational of our unpleasant options.  It is moral because it is rational in a world like ours.

     I would propose that game theory gives us considerable evidence that whatever conclusion we draw about inherent violent natures or inclinations of  human beings, one further argument is that violent conflict is simply put, rational—one might say unfortunately rational.  If it is rational to engage in war, perhaps even sometimes to initiate wars, then refraining from war is more likely to be morally supererogatory than obligatory.  A distinction here must be made between a rationality that is narrowly based on self-interest, and what I will call an interest in the general welfare—in others' or total welfare and in the future welfare of others.  It has been fairly easy to dismiss the possible rationality of short term self-interest as morally insignificant; it is much more difficult if an argument can be made that all or most others are better off in the long term if we engage in wars.  This is precisely what I will argue:  everyone is better off in the long term future if all parties have policies of initiating certain kinds of wars, namely retaliatory ones and even preemptive and preventive wars and refraining from others.  Furthermore this is a matter of game-theoretic and hence mathematical fact.  (There are some important assumptions and caveats in this conclusion.)  In particular, there are wars that would be clearly unjust by traditional Just War criteria but which nevertheless are in the long term general interest.

    One important class of games was almost from the beginning of game theory of considerable interest.  These are games that mirror the salient features of conflict between two parties, especially "non-cooperative" games.  In this general sense it does not matter whether the parties are individuals or nations.  The "conflicted" aspect of conflict games arise from the fact that what is advantageous to one party is to the disadvantage of the other.  Zero-sum games of course fall into this category.  Of special interest however are games that resemble the Prisoner's Dilemma.  If one party makes a move, say, initiating violence, and the other does not, then the first party comes out markedly ahead.  One further twist is necessary, namely that if both parties should initiate attacks, then both parties would be slightly worse off than if they both did not initiate this attack.  There are various ways of conceptualizing this scenario in terms of international conflict.  One can think of this as military expenditures.  It is fairly clear, for example, that each nation would be better off if it did not have military expenses because it did not need them—that is, if everyone would refrain from even having armed forces and thus lack the capacity to attack.

     Because of work by Nash and others—the so-called Nash equilibrium—it has plausibly been argued that rationality would dictate that the rational action for both parties is to make these expenditures and thus be slightly worse off than if they could both agree and be bound that neither party has military capacity.  From that position if either side would make a different move, it would be far worse off.   In the real world the problem is not just agreement but verification—adequate knowledge that the other party is indeed foregoing this capacity to attack.  The intuitive attractiveness and hence rationality of Nash equilibrium grows as the difference between the payoffs grow for an attacking or armed nation over one that does not counterattack or is unarmed.  It is not so important that we believe those taking part in this conflict are bloodthirsty and aggressive, or that they are greedy for the other's captured wealth.  Instead we might view such relatively frequent attacks, and military expenditures as the rational response to a  future threat.

     I have assumed up to now that we are looking at only at one opportunity to engage or not engage in conflict.  This is a description of the simple, one-play Prisoner's Dilemma (PD).  Of considerable more interest is the so-called Iterated Prisoner's Dilemma (IPD): here we engage in conflict over and over again.  This means that over time each party can gain some understanding of the other party's policies: one can "communicate" with the other party and can also persuade them to adopt certain policies through one's own policy.

     In computer simulations of the consequences of parties' adoptions of certain policies, Robert Axelrod has shown that a consistent superior strategy over almost all other strategies in IPD is Tit-for-Tat.  If your enemy attacked you on the last move, then you attack him on this move; otherwise refrain from aggression  He did this by staging a competition in which game theorists could submit what they thought was the most powerful strategy, and then played these submitted strategies against each other (with hundreds of iterations).

        Furthermore he gave a number of analyses of actual behavior—in international conflict and in legislative maneuvering—that seems to exhibit how rational parties would indeed react in something like the IPD.  In other words, he proposed that IPD was a useful model in understanding a wide variety of real-world activities in which parties sought to understand others' probable policies and to create optimal strategies themselves for dealing with most of the behavior they would encounter.  Later researchers have added hundreds of real-world scenarios that seem to duplicate these payoffs and in which parties slowly reach an equilibrium that approaches Tit-for-Tat.  (Related techniques have also been employed to model processes in evolutionary biology, in which successful strategies constitute adaptation to and environment.)

     Axelrod speculated that Tit-for-Tat had these advantages over other strategies because it had several features.  First, it was generous: it did not initiate attacks until attacked in a previous move.  Second it was clear:  an opponent could determine what strategy you were using.  Third it was universalizable: if every party adopted it they would be tolerably well off—at least in the sense of Nash Equilibrium.  Fourth, it is relatively forgiving: whatever the past or accumulated damage given you, you attack only if the opponent's last very move was an attack.   Fifth, it acts to compel others to adopt a similar strategy.

     It is important to recognize limitations on these results.  Axelrod's initial research was limited to two-party, symmetric conflicts.  The payoff matrix was fixed to four standard values.  Each party had perfect knowledge of the history of the conflict—what that party and its opponent had done last move or ten moves ago.  Measurement of success was performed using a range of strategies with a fairly limited grammar and there were no adaptive, learning strategies in which one party made serious attempts to learn the other party's strategy and adapt to it.  (Later work took account of the evolution of strategies that evolved according to success against other strategies.)  Finally, the measurement of success was crude, measuring only the relative success of one strategy over another.

     It is fairly obvious that these conditions are highly idealized when compared with real-world scenarios.  The simulations did not take into account the multi-national character of international conflict and the ability to form shifting alliances.  They did not mirror each nation's poor knowledge of history and of other nation's intentions or current strategy.  Finally, there is a huge array of means by which one nation can punish another for harming its interests, or threatening to harm those interests, from embargos and tariffs to massive nuclear attack.


Measuring Deep Rationality and Morality in IPD


In order to allow summative scoring of many iterations, and to make it so that a high total score indicates a successful strategy, Axelrod used the following payoff matrix: each party get 3 pts if both refrain from attacking ("cooperating" in traditional terminology); a party gets 5 points if it ambushes (attacks/"defects") the other party, while the ambushed party gets 0 points; and if both parties attack each other ("defect"), they each get 1 point.  The basic PD or IPD requires only that the ordering of these payoffs be structured in the way they are here, namely, 5> 3, 3>1, and 1>0, and not that their absolute values be fixed in any way.  The values could just as well be 500 for attacking a peaceful player (and 0 for the peaceful player), 499 for mutual peace, and 498 if both attack.  While these new values would preserve the essential elements of PD, they would create a strong incentive to avoid at almost all costs the victimization of the surprise attack.  Consequently it is far more informative to keep track of components for the total number of times one succeeds in attacking a peaceful player (AP), both being peaceful (PP), both attacking (AA) and are peaceful but attacked (PA).   The required ordering is AP>PP>AA>PA.  If both parties attack (AA), the result might be that we survive but have spent a great deal of money for an anti-ballistic missile system, while if we have no such system and are attacked the result could be our annihilation.

      A number of other modifications would seem to be necessary for minimal realism.  First, occasionally one is mistaken about whether the opponent really did attack on the last move (this is handled in game theory as "noise").  Occasionally one attacks mistakenly thinking an opponent attacked on the last move.  Second, if one is employing Tit-for-Tat and has any reason to suspect an opponent might be using a Tit-for-tat-like strategy (that was triggered by noise, or the opponent occasionally launches a probing attack), and for some reason the opponent has started attacking, then it would be useful to attempt to break out of a cycle of mutually attacking each other.  After n attacks, try one peaceful move and absorb the damage (a unilateral pause), hoping the opponent replies in kind.   Finally, scoring should distinguish between a "just," retaliatory attack and an attack that had its origins in other sources, such as in initiating an attack simply because one always attacks after an opponent has been peaceful for 5 moves. 

     Several observations about scoring.  First, notice that it might be rational to choose a strategy that is more likely to lose to some other strategies if the absolute damage from that strategy is less.  For example, in some cases one might achieve more victories but the damage to your own nation might be higher than it would be if you used a "losing" strategy.  Consider these scores:

Your strategy A against opponent:  you win, 50 to opponent's 45.

Your strategy C against opponent:  you lose, 100 to opponent's 120.

Here, strategy C is preferable to us, even though it loses to the opponent.  

      Secondly, morality would seem to require some attention to be paid in your calculations to what the likely expected total damage, to yourself and opponent, will be when using a certain strategy.  There is some, perhaps very small value that you would sacrifice in order to keep total damage low. Consider:

Your strategy D against opponent: you win, 50 to opponent's 45.

Your strategy E against opponent: you win, 51 to opponent's  -1000.

In other words, under strategy E the opponent suffers enormous losses (possibly innocent lives) to your slight gain, 51 to 50,  over the using strategy D.  In other words, particularly if the opponent's low score indicates losses of innocent lives, under some circumstances one should prefer winning strategies that do not entail such heavy losses for the opponent.

         Since in the popular (and perhaps distinctively Western-Christian) conception, retaliation and reprisal are almost always equated with atavistic revenge, we have what I would call tutored intuitions against all forms of retaliation.  These tutored intuitions about retaliation and deterrence have been proven, as an a priori matter, to be problematic by the mathematics of game theory.  With minimal assumptions about rationality,  strong rejection of retaliative punishment is a strategy that invariably produces a world in which there is more total destruction over worlds in which parties practice retaliation.

        In a series of computer simulations of the so-called Iterated (Two-player) Prisoner's Dilemma (IPD) with the modifications (and others) sketched above, I have taken standard models of conflict and applied them to preventive war.[2]  I have also incorporated some of the "moralizing" and non-summative features of scoring that are discussed above.  These programs allow two players to employ a wide variety of probabilistic strategies, including ones where one player or both are using ‘preventive war’ strategies.  In these strategies, one party occasionally has some fallible knowledge of the move of the opponent on the next move.  The strategy I used had a baseline probabilistic Tit-for-Tat strategy in which war was probabilistically initiated 1-15% of the time (termed ‘noise’), returned to non-attack mode (‘unilateral cease-fire’) after its own attack on the last move  5-15% of the time, and otherwise applied Tit-for-Tat.  The range of values for probabilities indicates that I had tournaments (of thousands of plays) in which each incremented value within the range was tried against all other combinations of parameters. The baseline preventive war strategy was one in which an attack was initiated 10-90% of the time if the opponent's next move (or two moves in the future) was an attack; however, this also included 1-20% of ‘false’ preventive attacks—i.e., preventive attacks where the opponent was in fact not attacking.  Otherwise, the strategy was as in the baseline Tit-for-Tat strategy.  One must remember that there is no single best strategy for IPD  simpliciter, particularly if both relative advantage and absolute measures of destruction are applied.  That is, one strategy may outperform another strategy, but the absolute destruction even to the winning player might be relatively high.  Even Tit-for-Tat can be bested by the Constant Attack strategy—as well as by other strategies.

     The results of applying tournaments to dozens of these strategies, each for thousands of iterations, are described as follows.  Preventive war strategies typically slightly outperformed baseline Tit-for-Tat strategies—roughly by blunting the effects of the enemy's attack or forcing them into ceasefires.   This was the case even when they always initiated some percentage of false preventive attacks which then lead to strings of retaliations against other tit-for-tat-like strategies.  The total destruction to both parties was, with certain values for preventive strategies, less than with any other similar retaliatory strategy without preventive war.  The preventive war strategies displayed a significant ‘threshold effect’, namely if mistaken prevention was much higher than 10% of the cases, then the total destruction to both players increased even if one player retained a relative advantage.  This threshold value of approximately a maximum of 10% mistaken preventive wars was highly sensitive to the absolute values in the payoff matrix and to the other probabilistic values in the strategy.

      Given the idealizations inherent in almost any game-theoretic description of a real-world conflict, this does not prove conclusively that some policy of preventive war is morally workable in actuality.  However, it does strongly suggest that the intuition or presumption against preventive war is, like the intuition against deterrent and retaliatory strategies, misguided.  Everyone's having such policies does not lead to more war than if no one had them—at least not in the general case in which opponents attack sometimes.  Perhaps more to the point, there is reason to suspect that these game-theoretic threshold effects (e.g., of sharply increasing total destruction for some values of mistaken preventive war) are deeply related to what would be a reasonable value for what I have elsewhere  called (JSCOPE 2005) "epistemic thresholds."  That is, preventive war is moral only if we have a strategy in which we believe the opponent will attack with some degree of reliability.  Our ability to determine these values by computationally intensive means indicates possible a priori constraints on epistemic thresholds, and thus that some universally adopted policies of preventive war are not only morally permissible but are morally preferable to their non-preventive variants.             




I cannot possibly investigate all the ways in which game theory might be relevant to the morality of war—jus in bello as well as jus ad bellum.  I hope I have indicated enough to show that some aspects of game theory might be relevant to moral considerations.  Observe that my calculus for game theory is different from the traditional cost-benefit analysis for one party since it will explicitly tally total destruction to all parties as a negative feature, as well as specially tallying unprovoked attacks.  In general, there are two crucial assumptions one must make in order for game theory to be relevant to moral theory about war in this way.  First, one must assume that whatever moral theory generally requires of us, it is constrained by rationality.  This rationality places some value on the reasonably expected long-term well-being of all parties.  It need not be a thorough-going consequentialism but is constrained by some consequentialist considerations.  As part of rationality we are assuming the following.  We are assuming that we must form a moral strategy based upon the assumption that the other parties behave in certain ways.  They, and we , will likely respond in certain ways to certain actions, especially when conceived by them as part of our consistent strategy.  Notable among such motives is some long-term self-interest.  This need not be a cynical Hobbesian picture that humans act only according to such motives, but the historical evidence is that they often do and certainly sometimes do.  These assumptions amount to two hypotheses for the philosophy of war: first, states that are parties to any agreement, demonstrated pattern of past behavior, or professed moral constraint will sometimes (maybe often) break free from this past pattern or promise if self-interest is sufficient.  Second, we assume that parties will recognize retaliation for what it is and not merely as our own spontaneous effort to advance self-interest.  Indeed one of the conclusions of Auman and Schelling is that it is rational in the long-term for parties to dish out punishment that is damaging to everyone in the short run.[3]  Promoting such punishment among all parties against defectors aids the identification of narrowly retaliative strategies.  

      A second assumption is that this incorporation of game theory via its connection to rationality implicitly endorses a "rule" as opposed to an act-based conception of morality.  Namely, it is strategies that become the proper object of moral assessment and not individual actions.  I do not make much of an apology for this, since I believe that some forms of rule utilitarianism (or rule egoism) seem less indefensible to us than they did a decade or so ago.[4]   There is of course no problem at all with non-consequentialist theories, since they are typically rule-based.  However, there is also a key difference between moral theory for individuals and that for states.  It is rarely an important factor in  the calculation of consequences for one individual to keep track of the probable life-strategies of every other individual.  One's behavior and perceived strategy does not have an effect on many others.  There are too many other individuals and most of your life is too hidden from me to infer a pattern even if we are acquaintances.  States, at least relatively large states, are different in these respects.  Histories of nations are accessible in ways that the histories of different individuals are not.  Hence it is possible for a leader to infer the probable strategy of another nation.  Furthermore, especially in the post-industrial age it is possible for a nation to punish, and be seen as punishing according to an inferred strategy, any other nation in a way that is utterly unthinkable for billions of individuals.  (Observe that this makes visible retaliation especially incumbent on larger states and alliances.)  In game-theoretic terms, there is closer to complete knowledge of the past behavior of states and to make at least crude inductions about future behavior.  If a rule-based moral theory is plausible for individuals, it is considerably more plausible for larger states.[5]

       The strategy, and the recognition by others of our consistent application of this strategy, is more important than any effects that may seem cruel and unnecessary in a given case.  In terms of Just War theory, war damage as punishment may thus fail both Chance of Success  and Proportionality with respect to that conflict; what may be more important is other nations' observations of the ruthless application of our strategy.  This is a well-known aspect of Geopolitical Realism that is typically maligned in moral-philosophical circles.  If game theory is correct, then we should also punish those who themselves fail to punish flagrant malefactors.  This does not meet the traditional criterion of Just Cause.  As we have already seen, game theory suggests we should accept some preventive strategies that may also violate Just Cause.

There are some other oddities that a game-theoretic approach to the morality of war will have—at least when compared with Just War theory.  For example, we might think that proper strategies should always have all parameters public: we tell everyone what they are and others can see from our actions that we are conforming to them.  This is indeed the case for the basic Tit-for-Tat strategy.  However, let us suppose that we make public our algorithmic criteria that would be necessary for us to initiate preventive war: n amount of evidence of m amount of opponent's weaponry.  That would unfortunately invite an opponent to work especially hard to hide exactly the kinds or amounts of evidence n, and it invites all opponents to accumulate weapons—up to just below m.  Consequently, with regard to some parameters, it is important not to be consistent and essentially to randomize our actions to some degree.  This will serve to obscure these parameters, and produce a less destructive world.  (This result is widely understood in strategic game theory, but it is easy to decry in the public arena.)

     Whither traditional Just War theory?  I am not sure I am ready to throw it out just yet, and certainly not ready to throw it all out.  In fact, some aspects of a sophisticated strategy (such as Proportionality) would merely translate the Just War component into some game-theoretic element of a wise and successful strategy.   Indeed, Tit-for-Tat—if we look at the Just Cause condition and not Success and Proportionality—is a formulation that accords with Just War theory in a majority of cases.  We might even view Just War theory as a "folk," or somewhat amateur but helpful attempt to capture the game-theoretic strategy that would be best for all.  Just War theory might even come close to describing some equilibrium or plateau: it is just not the optimal equilibrium we can reasonably hope to attain.  Certainly it would be good (indeed perfect) if every state would adopt it.   As it now stands, however it lacks three features.  First, it lacks epistemic criteria: the extent to which we should have evidence that a given condition is met.  This I have discussed elsewhere.  Second, it lacks a feature of basic Tit-for-Tat, namely its public clarity.  It would be difficult to determine if another nation were following Proportionality and Chance of Success, for example, although Just Cause would presumably be more transparent.  Finally, and most grievously, it lacks the compelling, enforcing character of the Tit-for-Tat variants I have been discussing.  It does not contain within itself a feature that promotes compliance with it.  It does not advocate punishing to some degree those who tolerate others' noncompliance.  It does not advocate possibly punishing noncompliance with more harm that the attacking nation caused.   It rejects producing more harm to everyone than simply ignoring an enemy's attack would produce.   Here we see the damage that the preoccupation with an act-moralistic approach produces over a game-theoretic approach.  We avoid short term harm to all parties from this event when there is now strong mathematical evidence that we are then producing more total long term harm.      

     Whither Geopolitical Realism?  Philosophers: sit down to dinner with the devil, and listen.





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[1] See the summary of their work accompanying the Nobel Prize at http://nobelprize.org/economics/laureates/2005/ecoadv05.pdf.

[2] They were written in the computer language Prolog over 2003-2006.  Each strategy was tested for runs of 500-1000 iterations (compared with Axelrod's original 400) and then these runs were repeated up to ten times and averaged so as to eliminate artifacts of the random-number generator.  Strategies were compared not just for relative advantage over other strategies, but also for total net destruction to both parties, a rough measure of the Just War criterion of Proportionality.  My purpose was originally to calculate a possible value for the "epistemic threshold" discussed in Dipert 2006.  

[3] Specifically that in some repeated games benefits to every party in a multi-party conflict will fall below what is otherwise guaranteed by Nash Equilibrium.

[4] See the excellent article on "Rule Consequentialism" in the Stanford Internet Encyclopedia of Philosophy.

[5] There are other, metaphysical reasons why a rule-based theory is preferable to an action-based one.  Act-utilitarianism for example runs into the calculation problem in part because it is examining consequences of an action conceived of merely as an event in the physical world.  However, actions are not just events.  To be an action, an event must minimally be the product of deliberation; this deliberation must involve the consideration of rules by the actor and the conclusion of the deliberation must be "guided by" (but not determined only by) some normative rules (i.e., what are normally called desires in the belief-desire model of action).   An action is thus ipso facto rule-guided and any moral flaw in the act must harken back to a flaw in the rule.  (Faulty beliefs, if one is culpable for them, can only have been produced by faulty normative belief-forming rules.  Thus again we arrive at rules.)