Apocalypse Mathematics: Game Theory and the Caribbean Nuclear Crisis

Move theory

“We played staring eyes, and, in my opinion, the enemy blinked,” said US Secretary of State Dean Rusk at the peak of the Cuban missile crisis in October 1962. He was referring to the signals that the Soviet Union gave, wanting to resolve the most dangerous nuclear confrontation between the two superpowers, which many analysts interpreted as a classic example of playing nuclear "chicken" (in Russian, the analogue of this game is called "hawks and pigeons").

The game of "chicken" is usually used to simulate conflicts in which each of the players headed for a collision. Players can be drivers approaching each other on a narrow road, each of which has a choice - to turn to avoid collisions, or not to turn off. In the story “Rebel Without a Cause” , which was later converted into a film with the participation of James Dean, the drivers were two teenagers, but they did not drive at each other, but toward the edge. The goal of the game was to not press the brakes first and thus turn into a “chicken”, and at the same time not to fall off a cliff.

Although the Caribbean missile crisis looks like a chicken game, in reality it is poorly modeled by this game. Another game more accurately describes the actions of the leaders of the United States and the Soviet Union, but even for this game, standard game theory does not fully describe the options available to them.

On the other hand, it reproduces or predicts the past of actions of the leaders of the theory of moves (theory of moves), based on the theory of games, but radically changing the standard rules of the game. More importantly, this theory sheds light on the dynamics of the game, based on the assumption that players think not only about the immediate consequences of their actions, but also about their influence on the game in the future.

I use the Caribbean nuclear crisis to illustrate certain parts of this theory, which is not just an abstract mathematical model, but also reflects the choice made in real life, the thought processes that led to it, and also explains the actions of living flesh and blood players. Theodor Sorensen, a special adviser to President John F. Kennedy, actually used the terminology of "moves" to describe the discussions of the executive committee (Excom, Executive Committee) of Kennedy's main advisers during the Cuban missile crisis:

"We discussed the reaction of the Soviets to any possible moves of the United States, our reaction to these actions of the Soviets, and so on, trying to go along each of these paths to a logical conclusion."

Classic game theory and nuclear crisis

Game theory is a branch of mathematics that studies decision making in social interactions. It applies to situations ( games ) in which two or more people (called players ) choose from two or more modes of action (called strategies ). The possible outcomes of the game depend on the actions chosen by all players, and can be evaluated in order of preference for each player.

In some games with two players and two strategies, there are player strategies that are in some sense “stable”. This is true when no player, deviating from his strategy, can achieve the best results. These two strategies together are called the Nash equilibrium, in honor of mathematician John Nash , who won the Nobel Prize in economics in 1994 for his work in the field of game theory. Nash equilibria do not necessarily lead to the best results for one or even two players. Moreover, in games that can be analyzed and where players can set only the rank of results (“ordinal games”), but cannot associate numerical values ​​with them (“cardinal games”) - they may not exist. (Although, as Nash showed, they always exist in cardinal games, but the Nash equilibrium in such games may include “mixed strategies”, which I will discuss below.)

The Cuban nuclear crisis was initiated by an attempt by the Soviet Union in October 1962 to install in Cuba medium-range and intermediate-range nuclear ballistic missiles capable of striking a large part of the United States. The goal of the United States was the immediate movement of Soviet missiles, and to achieve it, the top leadership of the United States seriously considered two strategies [ see Figure 1 ]:

1. The naval blockade (B) , or, as it was covertly called, "quarantine", to prevent the delivery of new missiles, followed by potentially more serious action that would force the Soviet Union to remove the already installed missiles.
2. A “surgical” airstrike (A) to destroy already installed missiles, as far as possible, which could potentially be followed by an invasion of the island.

The following alternatives were open to the leadership of the Soviet Union:

1. Review (W) of their missiles.
2. Conservation (M) of rockets on the island.

		Soviet Union (USSR)
		Review (W)	Preservation (M)
United States (USA)	Blockade (B)	Compromise (3.3)	Victory of the Soviets, defeat of the USA (2.4)
	Air strike (A)	US victory, defeat of the Soviets (4.2)	Nuclear War (1.1)

Figure 1: Cuban nuclear crisis as a chicken game

Key: (x, y) = (US win, Soviets win): 4 = best; 3 = slightly worse than best; 2 = slightly better than the worst; 1 = worst. Nash equilibrium is underlined.

These strategies can be considered alternative action programs that can be chosen by two parties, or “players” in the terminology of game theory. They lead to four possible outcomes that players must rank for as follows: 4 = best; 3 = slightly worse than best; 2 = slightly better than the worst; 1 = worst. That is, the greater the number, the greater the gain; but the winnings are only ordinal , that is, they denote only the order of winnings from best to worst, but not to the extent that the player prefers one result to another. The first number in each of the paired results is the player’s horizontal gain (US), the second number is the vertical gain of the player (USSR).

Needless to say, the election of the strategy, the likely results and the associated wins, shown in Figure 1, provide only a general framework of the picture of the crisis unfolding over thirteen days. Both sides considered more than two alternatives from the list, and each of them had several variations. The Soviets, for example, demanded the withdrawal of American missiles from Turkey as a quid pro quo for recalling their own missiles from Cuba. This requirement has been publicly ignored by the United States.

Nevertheless, most observers of this crisis believed that the two superpowers headed for a collision, which gave the name of one of the books devoted to this nuclear confrontation. In addition, they agree that none of the parties sought to take any irreparable steps, like one of the drivers playing in the “chicken”, defiantly disrupting the steering wheel of his car in view of the other driver, thus excluding the possibility of turning.

Although in a sense, the United States "won" by forcing the Soviets to withdraw their missiles, USSR First Secretary Nikita Khrushchev at the same time fished President Kennedy out of a promise not to attack Cuba, so this end result can be considered a kind of compromise. But for the game of "chicken" this is not a prediction on game theory, because the strategies associated with the compromise do not constitute Nash equilibrium.

To make sure of this, suppose that the game is in a compromise position (3.3), that is, the USA is blocking Cuba, and the USSR is withdrawing its missiles. This strategy is unstable, since both players have an incentive to reject to their more militant strategy. If the United States had deviated, changing its strategy to an airstrike, the game would have shifted to (4.2), improving the gains received by the United States; if the USSR had deviated, changing the strategy for the preservation of rockets, the game would have shifted to (2.4), giving the USSR a gain of 4. (This classic game theory scheme does not give us any information about which result will be chosen, because the winning table symmetric for both players. This is a frequent problem of interpreting the results of theoretical analysis of games where several equilibrium positions can arise.) Finally, if players get the mutually worst result (1.1), that is, nuclear war, then it is obvious that both will want to deviate from him that with makes related strategies, for example (3.3), unstable.

Move Theory and Nuclear Crisis

Using the chicken game to simulate a situation like the Caribbean crisis is problematic, not only because the compromise result (3.3) is unstable, but also because in the real world the two sides do not choose their strategies simultaneously or independently of each other, as it is supposed in the chicken game described above. The Soviets responded specifically to the blockade, after it was declared by the United States. Moreover, the fact that the United States considered the possibility of an escalation of the conflict, at least up to airstrike, suggests that the initial decision on the blockade was not considered final. That is, after the declaration of the blockade, the US still considered possible options for choosing a strategy.

Consequently, this game is better modeled as successive negotiations in which neither side made the “all or nothing” choice; both considered alternatives, in particular in the event that the opposite side did not respond in a way that the other side deemed appropriate. In the most serious deterioration of the nuclear deterrence relationship between the superpowers, which has been preserved since the Second World War, each of the parties cautiously probed their way, making threatening steps. The Soviet Union, fearing a US invasion of Cuba before the crisis, and also trying to maintain its strategic position in the world, concluded that the risk of installing missiles on the island was worth it. He believed that the United States, faced with a fait accompli (with a fait accompli), would refrain from attacking Cuba and not dare to take other harsh retaliatory measures. Even if the installation of missiles triggers a crisis, the Soviets did not consider the likelihood of war to be high (during the crisis, President Kennedy estimated the likelihood of war in the interval from 1/3 to 1/2), that is, the risk of provoking the United States would be rational for them.

There are valid reasons to believe that the top US leadership did not view confrontation as a chicken game, at least in the way it interpreted and ranked the possible results. I propose an alternative model of the Caribbean nuclear crisis in the form of a game, which I will call “Alternative” . In it, I will keep the same strategies of the players as in the “chicken,” but I will assume a different ranking and interpretation of the results by the United States [ see Figure 2 ]. Such rankings and interpretations better correspond to historical documents than the parameters of the “chicken” game, as can be judged by statements made by President Kennedy and the United States Air Force, as well as by the type and quantity of nuclear weapons available to the USSR (more on this below ).

1. BW : the choice of the US blockade and the recall of missiles by the Soviet Union is still considered a compromise for both players - (3.3).
2. BM : in the face of the blockade of the United States, the preservation by the Soviets of missiles in Cuba leads to the victory of the USSR (the best result for it) and the surrender of the US (the worst result for them) - (1.4).
3. AM : an airstrike that destroys the missiles saved by the Soviet Union is considered an “honorable” for the USA action (the best result for them) and the defeat of the Soviets (their worst result) - (4.1).
4. AW : an airstrike that destroys the missiles recalled by the Soviets is considered a “shameful” US action (the result is slightly better than the worst for them) and the defeat of the Councils (the result is slightly better than the worst) - (2.2).

		Soviet Union (USSR)
		Review (W)		Preservation (M)
United States (USA)	Blockade (B)	Compromise (3.3)	→	Soviet Victory, US capitulation (1.4)
	Air strike (A)	"Shameful" US action, defeat of the Soviets (2.2)	←	"Honorary" action of the USA, defeat of the Soviets (4.1)

Figure 2: The Caribbean Nuclear Crisis as an “Alternative”

Key: (x, y) = (win for the USA, win for the USSR), 4 = best; 3 = slightly worse than best; 2 = slightly better than the worst; 1 = worst. Non-myopic equilibria are bold. The arrows indicate the direction of the cycle.

Even though an air strike in both cases leads to the defeat of the Soviets, (2.2) and (4.1), I interpret (2.2) as causing the least damage to the USSR, because from the point of view of the rest of the world, an air strike can be considered as a blatantly overreaction, and therefore a “disgraceful” US action in the event of the existence of clear evidence that the Soviets are in the process of recalling missiles. On the other hand, in the absence of such evidence, an airstrike by the United States, which might have been followed by an invasion, would have been an action to oust Soviet missiles.

Statements from the top US leadership confirm compliance with "Alternative." In response to a letter from Khrushchev, Kennedy reports:

“If you agree to the dismantling of these weapons systems from Cuba ... we, for our part, will agree ... (a) to urgently remove the quarantine measures currently in force and (b) guarantee non-aggression in Cuba”,

which corresponds to the "Alternative", since (3.3) for the United States is preferable to (2.2), while (4.2) in the "chicken" is not preferable (3.3).

If the Soviets had kept their missiles, the United States would have preferred the blockade of an airstrike. According to Robert Kennedy, a close adviser to his brother at the time,

"If they do not remove these bases, we will remove them",

which corresponds to the "Alternative", since the US will prefer the result (4.1) to the result (1.4), rather than the result (1.1) to the result (2.4) in the game of "chicken."

Finally, it was well known that many advisers to President Kennedy were very reluctant to consider initiating an attack on Cuba, without exhausting less militant methods of action that could lead to the recall of missiles with less risk and greater compliance with America’s ideals and values. In particular, Robert Kennedy said that an immediate attack would look like “Pearl Harbor, on the contrary, and it would blacken the name of the United States in the pages of history”, which corresponds to “Alternative” because the US ranked AW a little better than the worst result (2 ) - as the “shameful” action of the States, and not as the best (4) - the victory of the USA - in the “chicken”.

Although Alternativa provides a more realistic view of the perception of the game participants than the chicken, the standard game theory almost does not help in explaining how the compromise was achieved and why it turned out to be a stable compromise (3.3). As in the chicken, the strategies associated with this result are not the Nash equilibrium, because the Soviets have an immediate incentive to move from (3.3) to (1.4).

However, unlike “chicken”, in “Alternative” there are no results at all that are Nash equilibria, with the exception of “mixed strategies”. These are strategies in which players randomize their chosen actions, choosing each of their two so-called pure strategies with given probabilities. But for the analysis of "Alternatives" it is impossible to use mixed strategies, because to perform such an analysis, each result must be linked to numerical gains, and not ranked in order.

The instability of the results in "Alternative" is best seen when studying the cycle of preferences, indicated by arrows, going in this game in a clockwise direction. Following these arrows means that this game is cyclical , and one player always has an immediate incentive to deviate from each state: the Soviets from (3.3) to (1.4); in the USA - from (1.4) to (4.1); for the Soviets, from (4.1) to (2.2); and in the United States, from (2.2) to (3.3). We again got indefinable, but not because of the presence of several Nash equilibria, as in the “chicken”, but because in “Alternative” there are no equilibria between pure strategies.

The rules of the game in the theory of moves

Then how do we explain the choice (3.3) in the "Alternative", and at the same time in the "chicken", given the non-equilibrium state of the standard game theory? It turns out that (3.3) is in both games a “non-myopic equilibrium” (nonmyopic equilibrium), and in “Alternative”, according to the theory of moves (theory of moves) (TOM) is the only such equilibrium. By postulating that the players think ahead not only for the immediate effects of the moves, but also for the consequences of counter-moves in response to these moves, counter-moves and so on, TOM extends the strategic analysis of the conflict to a more distant future.

Of course, the theory of games allows you to take into account such thinking through the analysis of "game trees", which describe the sequential actions of players over time. But the game tree is constantly changing every time a crisis develops. In contrast to this, in the “Alternative” the configuration of the winnings remains more or less constant, although there the players are in a modified matrix. In essence, TOM, describing wins in a single game, but allowing players to do consecutive calculations of moves in different positions, adds non-myopic thinking to the proposed classical theory of gaming economics.

The founders of game theory, John von Neumann and Oscar Morgenstern, defined the game as “a set of rules describing it”. Although the TOM rules apply to all games between two players, here I will assume that each player has only two strategies. Four rules of the TOM game describe the possible choices of players at each stage of the game:

Rules of the game

1. The game begins with the initial state given by the intersection of the row and column in the matrix of winnings.
2. Any player can unilaterally change his strategy, that is, make a move, and thus transfer the initial state to a new state in the same row or column as the initial state. The strategy-changing player is called player 1 ( P1 ).
3. Player 2 ( P2 ) can answer, =, unilaterally changing his strategy, thus transferring the game to a new state.
4. Answers continue to alternate as long as the player ( P1 or P2 ), who must go next, does not change his strategy. When this happens, the game ends in its final state, which is the result of the game.

End rule

1. A player does not move from the initial state if his moves (i) lead to a less preferred outcome, or (ii) return the game to the initial state, making this state result.

Advantage rule

1. If for one player it is rational to move, and for another - not to move from the initial state, then the move takes precedence: it cancels the stay in place, so the result will be caused by the player who made the move.

Notice that the sequence of moves and counter moves strictly alternates: let's say, the player first walks horizontally, then the player vertically, and so on, until one of the players stops, and at this stage the state will be final, which means the result of the game. I assume that the players gain does not accumulate when they are in the state, unless it becomes the result of the game (which can also be the initial state, if the players decide not to move out of it).

To assume the opposite, it is necessary that the winnings are numerical, and not just ranked, then players could accumulate them, passing through the state. But in many games of the real world, wins are difficult to quantify or summarize according to the states in which they were located. Moreover, in many games a great reward is extremely dependent on the final state reached, and not on how it was achieved. In politics, for example, winning for most politicians is not in campaigning, because they are laborious and costly, but in victory.

Rule 1 is very different from the corresponding game rule in standard game theory, where players simultaneously select strategies from the matrix game that determines its result. Instead of starting with the choice of strategy, TOM assumes that at the beginning of the game, players are already in some state and receive a gain from this state only if they remain in it . Based on these winnings, they individually have to decide whether to change this state, trying to achieve the best.

Of course, some decisions are made collectively by the players, and in this case it is reasonable to say that they choose strategies from scratch, or at the same time, or by coordinating their actions. But if, say, two countries coordinate their actions, for example, they agree to sign a contract, then an important strategic question is what individual calculations led them to this situation. The formal act of jointly signing a treaty is the culmination of their negotiations and does not reveal the process of counter-moves preceding this signing. It is for the disclosure of these negotiations and the underlying calculations that TOM is intended.

Let's continue this example: the parties signing the contract were in a certain previous state, from which both decided to move - or, probably, only one decided to move, and the other could not hinder this move (advantage rule). Over time, they fell into a new state, after, let's say, negotiations on signing, and in this state it is rational for both countries to sign a previously negotiated agreement.

As is the case with the signing of the contract, almost all the results of the observed games have their own history. TOM seeks to strategically explain the development of (temporary) states, which has led to led to a (more permanent) result. Consequently, the game begins in the initial state, in which players receive winnings only if they remain in this state and it becomes the final state, or the result, of the game.

If they do not remain in this state, they still know what the winnings would have been if they had remained in the state; therefore, they can make a rational calculation of the benefits of maintaining a state or moving out of it. They move precisely because they have calculated that they can improve the situation by changing the strategy, waiting for the best result, when the process of moves and counter-moves finally comes to an end. When the game starts in a different state, the game will be different, but the configuration of the winnings will remain the same.

Rules 1 - 4 (rules of the game) do not say anything about what causes the game to end, but only about when it ends: completion occurs when "the player who must go next does not change his strategy" (rule 4). Но когда рационально будет не продолжать движение, или вообще не перемещаться из исходного состояния?

Правило завершения гласит, что это происходит, когда игрок не перемещается из исходного состояния. Условие (i) объяснений не требует, но условие (ii) нужно обосновать. Оно гласит, что если после хода P1 для партии игры будет рационально вернуться по циклу к исходному состоянию, то P1 не будет перемещаться. В конце концов, в чём смысл затевать весь процесс ходов-контрходов, если партия просто вернётся на «первую клетку поля», с учётом того, что по пути к результату игроки не получат никаких выигрышей?

Обратная индукция

Чтобы определить, на чём завершится партия, когда хотя бы один игрок захочет переместиться из исходного состояния, я предположу, что игроки используют обратную индукцию . Это процесс рассуждения, в котором игроки, проходя назад от последнего возможного хода игры, предвидят рациональные действия друг друга. Для этого я предположу, что каждый из них имеет полную информацию о предпочтениях другого, поэтому каждый может вычислить рациональные действия другого игрока, а также свои собственные, относительно решения о том, нужно ли перемещаться из исходного состояния, или любого последующего состояния.

Чтобы проиллюстрировать обратную индукцию, давайте снова рассмотрим игру «Альтернатива» на рисунке 2. После обнаружения ракет и наложения Штатами блокады игра находилась в состоянии BM, наихудшем для США (1) и наилучшем для Советского Союза (4). Теперь рассмотрим развитие ходов по часовой стрелке, которое могут инициировать США, переместившись в AM, после чего СССР переходит в AW, и так далее, предполагая, что игроки могут предугадать вероятность того, что игра совершит один полный цикл и вернётся к исходному состоянию (состоянию 1):

	Состояние 1		Состояние 2		Состояние 3		Состояние 4		Состояние 1
США начинают	США (1,4)	→	СССР (4,1)	→	США (2,2)	→ |	СССР (3,3)	→	(1,4)
Выжившее	(2,2)		(2,2)		(2,2)		(1,4)

Это дерево игры, только нарисованное не вертикально, а горизонтально. Выжившее — это состояние, выбранное на каждом этапе в результате обратной индукции. Оно определяется возвратом из того состояния, в котором игра теоретически может завершиться (состояние 1, при завершении цикла).

Предположим, что чередующиеся ходы игроков в «Альтернативу» делались по часовой стрелке от (1,4) к (4,1), потом к (2,2) и к (3, 3), и на этом этапе СССР в состоянии 4 должен был решать, остановиться ли на (3,3) или завершить цикл, вернувшись к (1,4). Очевидно, что СССР предпочтёт результат (1,4) результату (3,3), поэтому (1,4) указан как выжившее состояние под (3,3): так как СССР вернёт процесс обратно к (1,4), если он достигнет (3,3), то игроки знают, что если процесс ходов-контрходов достигнет этого состояния, то результатом будет (1,4).

Зная это, будут ли США в предыдущем состоянии (2,2) перемещаться в (3,3)? Так как США предпочтут (2,2) выжившему в (3,3) состоянию, а именно (1,4), ответом будет «нет». Следовательно (2,2) становится выжившим состоянием, когда США должны выбрать между остановкой в (2,2) и перемещением в (3,3) — что, как я только что показал, превратится в (1,4) после достижения (3,3).

В предыдущем состоянии (4,1) СССР предпочтёт переместиться к (2,2), а не останавливаться в (4,1), поэтому (2,2) снова будет выжившим, если процесс достигнет (4,1). Аналогично, в исходном состоянии (1,4), поскольку США предпочтут предыдущее выжившее состояние (2,2) состоянию (1,4), то в этом состоянии выжившим тоже будет (2,2).

Тот факт, что (2,2) является выжившим в исходном состоянии (1,4) означает, что для США рационально переместиться в (4,1), а СССР затем перейти в (2,2), где процесс остановится, делая (2,2) рациональным выбором, если США сделают первый ход из исходного состояния (1,4). То есть вернувшись обратно от выбора СССР о завершении или незавершении цикла из (3,3), игроки могут обратить процесс, и заглядывая вперёд, определить, что будет рационально сделать для каждого из них. Я указываю, что для процесса рационально остановиться на (2,2), поставив вертикальную черту, препятствующую исходящей из (2,2) стрелке, и подчеркнув на этом этапе (2,2).

Заметьте, что (2,2) в состоянии AM хуже для обоих игроков, чем (3,3) в состоянии BW. Может ли СССР, вместо того, чтобы позволить США инициировать процесс ходов-контрходов в состоянии (1,4), улучшить свою ситуацию, перехватив инициативу и двинувшись против часовой стрелки из своего наилучшего состояния (1,4)? Ответ положительный, более того, в интересах США также позволить СССР начать этот процесс, как это видно на следующем развитии ходов от (1,4) против часовой стрелки:

	Состояние 1		Состояние 2		Состояние 3		Состояние 4		Состояние 1
СССР начинает	СССР (1,4)	→	США (3,3)	→|	СССР (2,2)	→	США (4,1)	→	(1,4)
Выжившее	(3,3)		(3,3)		(2,2)		(4,1)

СССР, действуя «великодушно», перемещается из состояния победы BM (4) к компромиссу (3) в BW, и делает для США рациональным завершение игры в (3,3), что обозначено заблокированной стрелкой из состояния 2. Разумеется, именно это и произошло в кризисе, с угрозой дальнейшей эскалации со стороны США, в том числе вынужденного всплытия советских подводных лодок, а также авиаудара (ВВС США оценивали вероятность уничтожения всех ракет в 90%), став для Советов стимулом для отзыва всех своих ракет.

Применение TOM

Как и любая научная теория, расчёты TOM не могут принимать во внимание эмпирическую реальность ситуации. Например, во втором расчёте обратной индукции сложно представить перемещение Советского Союза из состояния 3 в состояние 4, включающее сохранение (через повторную установку?) ракет на Кубе после их отзыва и авиаудара. Однако, если переход в состояние 4, а позже обратно к состоянию 1 были исключены как невыполнимые, то результат был бы тем же: при выполнении обратной индукции в состоянии 3 для СССР будет рационально изначально переместиться в состояние 2 (компромисс), на котором игра остановится.

Компромисс также будет рациональным в первом расчёте обратной индукции, если тот же ход (возврат к сохранению ракет), который при этом развитии событий является переходом из состояния 4 в состояние 1, считается невыполнимым: выполняя обратную индукцию в состоянии 4, для США будет рационально продолжить эскалацию до авиаудара, чтобы вызвать ходы, приводящие игроков к компромиссу в состоянии 4. Так как для обеих сторон будет менее затратно, если Советский Союз станет инициатором компромисса, устраняя необходимость авиаудара, то неудивительно, что именно это и произошло.

Подведём итог: теория ходов превращает теорию игр в более динамичную теорию. Постулируя, что игроки продумывают наперёд не только ближайшие последствия ходов, но и последствия контрходов в ответ на эти ходы, контр-контрходов и так далее, она расширяет стратегически анализ конфликтов в более отдалённое будущее. TOM также использовалась для того, чтобы показать возможное влияние разных степеней применения силы (перемещений, приказов и угроз) на результаты конфликта, а также продемонстрировать то, как может воздействовать на выбор игроков дезинформация. Эти концепции и анализ проиллюстрированы множеством разных примеров, от конфликтов в Библии до современных споров и столкновений.

Дополнительное чтение

1. «Theory of Moves», Steven J. Brams. Cambridge University Press, 1994.
2. «Game Theory and Emotions», Steven J. Brams in Rationality and Society, Vol. 9, No. 1, pages 93-127, February 1997.
3. «Long-term Behaviour in the Theory of Moves», Stephen J. Willson, in Theory and Decision, Vol. 45, No. 3, pages 201-240, December 1998.
4. «Catch-22 and King-of-the-Mountain Games: Cycling, Frustration and Power», Steven J. Brams and Christopher B. Jones, in Rationality and Society , Vol. 11, No. 2, pages 139-167, May 1999.
5. «Modeling Free Choice in Games», Steven J. Brams in Topics in Game Theory and Mathematical Economics: Essays in Honor of Robert J. Aumann, pages 41-62. Edited by Myrna H. Wooders. American Mathematical Society, 1999.

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about the author

Стивен Дж. Брэмс (Steven J. Brams) — профессор политики в Нью-Йоркском университете. Он является автором и соавтором 13 книг по применению теории игр и теории социального выбора в голосованиях и выборах, переговорах и справедливости, международных отношениях, Библии и теологии. Его последние книги: Fair Division: From Cake-Cutting to Dispute Resolution (1996 год) и The Win-Win Solution: Guaranteeing Fair Shares to Everybody (1999 год) выпущены в соавторстве с Аланом Д. Тейлором. Он член Американской ассоциации развития науки, Общества «общественного выбора», стипендиат стипендии Гуггенхайма, приглашённый эксперт Фонда Рассела Сейджа и президент Международного Общества мирной науки.

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