Also known as the theory of games, the much talked about Game theory is essentially applies statistical logic to the choice of strategies. According to The American Heritage Dictionaries “Game theory is a mathematical method of decision-making in which a competitive situation is analyzed to determine the optimal course of action for an interested party”. Today, it finds its application in academic fields as diverse as biology, psychology, sociology, philosophy, economics, computers science and nuclear strategies. The game theory was developed by John Von Neumann and Oskar Morgenstern as a tool for understanding economic behavior. In the 1970s, game theory was also applied to animal behavior, including species’ development by natural selection. With interesting games like the prisoner’s dilemma, game theory has now found its place strongly in political science, ethics and philosophy. Game theory has recently drawn attention from computer scientists because of its use in artificial intelligence and cybernetics. In the year 1994, the Nobel Memorial Prize in Economic Sciences was awarded to Nash, John Harsanyi, and Reinhard Selten and in the year 2005 to Robert J. Aumann and Thomas C. Schelling (2005) for applying game theory to aspects of economics (nuclear non proliferation).
What is a game? In the game theory, games are referred as well-defined mathematical objects consisting of a set of players, a set of moves (or strategies) and a specification of payoffs for each combination of strategies. Thus, two to n no. of players or groups of players choose strategies designed to maximize their own winnings or to minimize their opponent’s winnings.
Player 2 chooses left
Player 2 chooses right
Player 1 chooses top
Player 1 chooses bottom
Normal form: In a normal form, it is presumed that each player acts simultaneously or without knowing the actions of the other. A normal game is in the form of a matrix which shows the players, strategies, and payoffs. Referring to the figure above, there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by Player 1; the second is for Player 2. If the player 1 plays top and that player 2 plays left. Then player 1 gets 4, and player 2 gets 3.
Extensive form: If the players have some information about the choices of other players, the game is presented in an extensive form. Here, games are presented as trees where each vertex (or node) represents a point of choice for a player. The player is identified by a number listed by the vertex and the lines out of the vertex represent a possible action for that player. The payoffs for both the players are specified at the bottom of the tree. In the figure above, Player 1 moves first and chooses either F or U. Player 2 sees Player 1’s move and then chooses A or R. Suppose that Player 1 chooses U and then Player 2 chooses A, then Player 1 gets 8 and Player 2 gets 2. The game can also be played by placing a dotted line or circle is drawn around two different vertices to represent them as being part of the same information set (i.e., the players do not know their points).
A Zero-Sum Game
The concept of zero sum and non-zero sum: In zero-sum games the total
benefit to all players in the game, for every combination of strategies, will always total to zero (or a player will benefit only at the expense of others). An example is Poker. On the other hand, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another. The most popular is prisoner’s dilemma where some outcomes have net results greater or less than zero.
Von Neumann (the developer of the game theory) and Morgenstern concentrated only on zero-sum games. However in the 1950’s, John F. Nash overcame this restriction by offering a distinction between cooperative and noncooperative games. In noncooperative games, no outside authority assures that players stick to the same predetermined rules, and binding agreements are not feasible. Also he highlighted that in noncooperative games there exist sets of optimal strategies (Nash equilibrium) used by the players which does not enables any player to benefit by unilaterally changing his or her strategy if the strategies of the others remain unchanged. Noncooperative games are known to be more common in the real world. Nash also suggested that the study of cooperative games can be done by their reduction to noncooperative form and proposed a “Nash program” for it. He also introduced the concept of bargaining in the game. Before we proceed to the applications of Game theory in the nuclear arena, we need to understand two important concepts
Nash Equilibrium: In a Nash Equilibrium, assume that (S, f) be a game, where S is the set of strategy profiles and f is the set of payoff profiles. When each player chooses strategy resulting in strategy profile x = (x1,…,xn) then player i obtains payoff fi(x). A strategy profile is a Nash equilibrium (NE) if no deviation in strategy by any single player is profitable, that is, if for all i
A game can have a pure strategy NE or a NE in its mixed extension (that of choosing a pure strategy stochastically with a fixed frequency). Nash proved that, if we allow mixed strategies (players choose strategies randomly according to pre-assigned probabilities), then every n-player game in which every player can choose from many strategies admits at least one Nash equilibrium.
Prisoner ’s Dilemma : This is another popular game where each player pursuing his own self-interest leads both players to be worse off than if had they not pursued their own self-interests. To explain, we suppose that two persons Dave and Henry are accused of fraudulent trading activities. Both are being interrogated separately and do not know what the other is saying. And both want to minimize the amount of time spent in jail. However, the dilemma is :
If Dave pleads not guilty and Henry confesses, Henry will receive the minimum sentence of one year, and Dave will have to stay in jail for the maximum sentence of five years.
If nobody makes any implications they will both receive a sentence of two years.
If both decide to plead guilty and implicate their partner, they will both receive a sentence of three years.
If Henry pleads not guilty and Dave confesses, Dave will receive the minimum sentence of one year, and Henry will have to stay in jail for the maximum five years.
Prisoner B Stays Silent
Prisoner B Betrays
Prisoner A Stays Silent
Both serve six months
Prisoner A serves ten years; Prisoner B goes free
Prisoner A Betrays
Prisoner B serves ten years; Prisoner A goes free
Both serve two years
Right choice would be to cooperate as this would reduce the total jail time served by them to one year.
Game Theory and Nuclear Proliferation:
Nuclear proliferation is basically the spread of nuclear technology including nuclear power plants but especially nuclear weapons from nation to nation. Nuclear proliferation across the globe is extremely dangerous as it may increase the possibility of nuclear warfare causing massive world destruction besides de-stabilizing international relations and endangering the national sovereignty of independent nations. Despite these threats many small countries have also been pursuing their own nuclear goals. Riding their reasons high on self defence, they question the very right of only few powerful countries owning such weapons. The idea to promote nuclear non-proliferation began only in the late-1960s, when five nations had acquired nuclear weapons (United States of America, Russia (formerly the Soviet Union), the United Kingdom, France, and the People’s Republic of China.). Since then, the primary focus of anti-proliferation efforts has been to maintain control over the specialized materials especially Uranium and Plutonium necessary to build such devices. However, the actual work started with the establishment of International Atomic Energy Agency (IAEA) in 1957 by the United Nations. IAEA organization has been the primary international anti-proliferation organization operating a safeguards system as specified under the Nuclear Non-Proliferation Treaty (NPT) of 1968. It has involved cooperation in developing nuclear energy while ensuring that civil uranium, plutonium, and associated plants are used only for peaceful non nuclear weapons purposes.
Current Nuclear Situation: India and Pakistan- two non-signatory states of the NPT have conducted nuclear tests. Israel is also strongly suspected to have an arsenal of nuclear weapons: there have been reports that over 100 nuclear weapons might be in its inventory. North Korea has publicly declared itself to possess nuclear weapons though it has not conducted any confirmed tests and its ultimate status is still unknown. And the latest, Iran has been accused by Western nations of attempting to develop uranium enrichment technology for weapons purposes. IAEA has thus referred Iran (signatory to the NPT) to the United Nations Security Council in response to their possible nuclear programs.
Contribution of Schelling: The very characteristic of a military struggle requires that one party or both parties will suffer losses, which need not involve losing goods to one another- thus it represents a non-zero-sum game. The same holds good for diplomatic and bargaining situations since both players may win. According to Yale, 2006, “Thomas C Schelling suggests that the two basic goals of military engagements are (i) to take away part or all of the enemy’s property (ii) to destroy the enemy or his property. Some approaches however circumvent the fact that the game is not zero-sum, by investigating a related zero-sum game. Thomas C Schelling’s work was instrumental in game theory, which became the dominant tool for analyzing the age-old question of why some groups, organizations and countries succeed in fostering cooperation, while others suffer from conflict.”2 In the late 1950s, Schelling, through his book The Strategy of Conflict, set forth his vision of the game theory at the backdrop of the growing nuclear arms race and observed that a nation could have long-term success by giving up some short-term advantages, even if that meant worsening its own options. He felt that by making concessions, the stronger party could build trust with the other party and that long-term relationship could be more beneficial to both. These insights are of great significance for conflict resolution and in building confidence steps in the hope of resolving conflict, example-in the Middle East. Schelling’s research has also led to new developments in game theory and its application in the realm of international politics concerning nuclear weapons- For instance, his analysis of strategic commitments has explained the delegation of political decision power. In nuclear deterrence, Schelling has explained why no nation would use a nuclear weapon because retribution would be assured. This would probably prevent nations such as Iran or North Korea from using nuclear weapons. Schelling’s co-laureate, Robert J. Aumann, was the first to analyze “infinitely repeated games”, which helped explain why some people or communities cooperate better than others over time. As a mathematician, Aumann’s contribution was to show how peaceful cooperation is often an equilibrium solution. Shelling finds that today’s situation as compared to the past cold war time is much more tense and complex. According to Washburn, 2006, “Shelling says that academics struggling to find solutions in a place like Iraq, for example, have to go beyond the confines of their offices. Theorists need a strong grasp of the facts on the ground and a deep understanding of the local culture before they can help. ”
On the other hand, Economist Gordon Hanson finds that today large countries and multinational corporations are disproportionately benefiting from the reduction in tariffs and other trade barriers. He says that this imbalance will threaten U.S. security. Pakistan, for example, doesn’t have an incentive to cooperate with the U.S. on security measures if small Pakistani firms can’t participate in the global economy.” Standard game theory works on the assumption that people have the same common understanding of the structure of the game, or common knowledge. But in reality, International political leaders often perceive the world in very different terms. For instance: Does George W. Bush have any clear idea of what Iran leaders are planning? Or were we able to fathom Saddam Hussein’s motives clearly? According to Tyler 2003, “ When we confront an opponent with nuclear weapons, we will misread cues, signals, threats, and responses, most of all when the opponent stands outside of Western culture. They will misread us in turn. We run the risk of unintended escalation from deluded sets of leaders, noting that you need only one side to make a fatal mistake. The more countries have nuclear weapons, the more likely is such a mistake to happen, and we haven’t even considered the problem of non-deterrable terrorists.”
According to Bracken, 2003, the core features of the current nuclear environment with respect to the game theory are:
1. An n-player game or a multiplayer game. Game theory tells us that even three player duels create great complexity. Equilibrium and stability are harder to achieve. Stability requires more agreement and trust. Today, nuclear deterrence is far more complex. In the Cold War, the options in a potential nuclear engagement were basically wait or shoot. However, today, a state might wait while others destroy each other, preserving its arsenal to threaten or finish a weakened adversary. 2. Nuclear weapons and the state- Nuclear weapons have become an essential part of state-building programs. In Iraq, North Korea, Pakistan, India, and Israel, nuclear weapons are used to define and empower the state.
3. Asian roots- All today’s emergent nuclear powers are Asian states. North Korea seeks nuclear weapons for self-reliance and a search for respect, Pakistan used Islamic fundamentalism to extend its power into Afghanistan and Central Asia. These decisions are both dangerous and misguided. Nationalism drives absurd behavior and strange decisions.
4. The cost of defense- The new nuclear powers are generally poor. These states tend to run down their conventional forces to pay for nuclear weapons and missiles. This tendency creates a new kind of instability. The increased reliance on nuclear weapons without a complementary conventional deterrent increases crisis instability.
5. Second-mover advantage- Today’s nuclear weapons states can observe states that went nuclear in the past to find out what works. This advantage creates the likelihood that states will take great risks.
According to Zaino, 2002, “Robert Wright, author of Nonzero: The Logic Of Human Destiny And The Moral Animal, has explained how technology and globalization have fostered the spread of the nonzero-sum game, a game theory term that means that the fates of both parties in a relationship are positively correlated. He believes that collaboration is the key to success in the game. Globalization raises the nonzero-sum stakes. When nations’ economies are so tightly integrated with each other, both suffer during war, and both profit during peace. He also said that in a world in which terrorists can coordinate a hijacking plan via cell phones, it’s imperative that nations reach out to the people in the street who are terribly unhappy, because the unhappy they grow, the more danger they present to the world.”
According to John C. Harsanyi, et al.1992 the simplest solution concepts to nuclear proliferation are –
(1) Damage Limiting Approach- A player plays “defensively”, by playing his own payoff matrix as if it were the matrix of a zero-sum game. This may be a good approximation if the game is almost zero-sum. This methodology has two benefits-one doesn’t have to know the opponent’s payoff and secondly the approach protects the player from unpleasant surprises since her is already defending against the worst. However such an approach is short-sighted, unless the opponent really was vindictive. Workers in the field of strategic nuclear warfare call this the “Damage Limiting” approach.
(2)Assured Destruction Approach-The complementary possibility is for a player to play vindictive, and play ‘offensively”, so as to minimize the opponent’s payoff. Again, if the game is nearly zero-sum this will be reasonable. Otherwise, it will give stability if the opponent uses the “defensive” strategy described above. This is called as the “Assured Destruction” approach.
According to John C. Harsanyi, et al. 1992, “Each of the above policies provides a recommended strategy (perhaps unique), and an inequality on the value for one player. (If the D. L. strategy would result in a value of v 1 even with the world against me, it will result in ?v 1 anyway; and if the A.D. strategy would hold my opponent to a return of v 2 even when he attempts to maximize his payoff, it will hold him to ? v 2 whatever he does.”
The Damage-Limiting (D. L.) threat strategy is the solution to a minimax problem with our own payoff as the criterion while the Assured Destruction (A.D.) threat strategy is the solution to a minimax problem with the negative of the opponent’s payoff as the criterion. Any threat strategy is the result of solving a minimax. problem with some linear combination of the the two threats as the criterion; i.e , each possible threat-point must be a solution to a problem of the form:
max min (? . P I (t I, t II ) – (1-?).P II (t I ‘ t II )), where 0 ? ? ? 1.
t I t II
(Note that ? = I corresponds to the D. L. threat-point and ? = 0 corresponds to the A. D. threat-point strategy. Two conclusions are drawn as per John C. Harsanyi, et al. 1992 : “first the set of solutions to the spectrum of minimax problems described above includes all threats which could be prescribed by the Nash solution; and, second, the true payoffs (assuming a rational form of cooperation between the opponents, after the threats are: made) will lie on the curve of efficient payoffs, which in general will be far “better than” — i.e. . above and to the right of — the threat points. The actions on that threat curve correspond to actions which we should be prepared to take, and the selection of a specific threat from that complex must depend on the “efficient” upper – right boundary K’ of the attainable set K of payoffs.”
Today, we find the world leaders having a hot debate over the consequences of nuclear proliferation and the viability of deterrence in the newly-nuclear states. And this issue finds the research community into two opposed and mutually exclusive groups. Kraig, 2006 finds that “One set of scholars believes in the efficacy of deterrence and thus tends to favor proliferation, while another loose grouping of research efforts questions the viability of deterrence in the developing countries. The former group has generally pitched its arguments at an abstract level (primarily through non-formal research), while the latter group has based its critiques on specific countries or on subtopics within the theory of deterrence (such as the fragility of command and control systems).” And in game theory as John C. Harsanyi, et al. 1992 states “neither Nash’s solution nor any other mathematical artifact can always provide a reasonable decision on what to do; in addition to the troubles mentioned above, the definition is static, and it is not at all obvious how one should apply it to a dynamic and fluctuating situation, nor how one should consider the problem of recurrent threats. Nevertheless this concept of threat presents a way of formulating and thinking about certain problems which appear intractable without it; and this is the main contribution which any theory can make.”
 John C. Harsanyi, John P. Mayberry and Herbert E. Scarf, eds. 1992. Game-Theoretic Models of Cooperation and Conflict: Westview Press- 65, 66.
 Kraig, Michael R. 2006. Nuclear Deterrence in the Developing World: A Game-Theoretic Treatment by <http://jpr.sagepub.com/cgi/content/abstract/36/2/141> (accessed 10 May 2006)
 Zaino, Jennifer. 18 March 2002. Spring Conference: Playing the Nonzero-Sum Game
< http://www.informationweek.com/story/IWK20020318S0003> (accessed 9 May 2006)
 Tyler, Cowen. 25 September 2003. The Game Theory of Nuclear Proliferation< http://www.tcsdaily.com/article.aspx?id=092503A (accessed 9 May 2006)
 Bracken, Paul. 5 November 2003. The Structure of the Second Nuclear Age by Professor Paul
< http://web.mit.edu/ssp/seminars/wed_archives_03fall/bracken.htm> (accessed 9 May 2006)
 Washburn, David. 22 March 2006. Nobel winner, others discuss world conflicts and cooperation in La Jolla < http://www.signonsandiego.com/news/business/20060322-9999-1b22schellin.html >
 31 march 2006. Events to explore legacy of Hiroshima, nuclear proliferation
< http://www.yale.edu/opa/v34.n24/story10.html > (accessed 9 May 2006)
 The American Heritage Dictionaries < http://www.answers.com/game%20theory > (accessed 7 may 2006)