ECOS3012 Creative Project:
Strategic Interaction in the Cold War
Introduction
Following the defeat of Hitler in World War II, tensions resurfaced between the Soviet Union
(USSR) and the United States of America (USA), arguably the two most powerful world states at
the time. Spurred by intense mutual distrust, by the 1950s, both sides had accumulated vast
arsenals of nuclear weapons. Why, then, did we never observe any direct conflict between the
USA and USSR during the Cold War? More importantly, is it possible that the accumulation of
nuclear weapons discouraged, rather than promoted, outright conflict? We aim to answer these
questions by modelling the Cold War as an infinitely repeated game with certainty.
Evidence and modelling assumptions
We describe each element of the game in turn, justifying our assumptions using evidence.
Firstly, we assume that the only two players are the USA and the USSR. Although the Cold War
involved other countries as allies, the USA and USSR were the two major opposing sides. Both
countries emerged as “superpowers” from the defeat of Hitler in World War II, and their
ideologies were in conflict. During the Cold War, both competed for political and ideological
influence throughout the world, as described by Green (2013) (link to video). Overy (2010)
describes this rivalry:
The roots of the Cold War lay in the Russian revolution of 1917. Communism, with its belief
in its own inevitable domination of the world, was seen as a profound threat to the world
capitalist system, of which the USA was taking the leadership.
These ideological tensions were also captured by many propaganda campaigns, as evidenced by
the posters below.
Figure 1: (a) American propaganda poster (1947), produced by the Catholic Catechetical Guild
Educational Society. This was part of a pamphlet distributed in the US which aimed to raise
fears about a communist takeover.
(a) (b)
(b) Soviet propaganda poster (November 1930), produced by Viktor Deni, entitled “Comrade
Lenin Cleanses the Earth of Filth”. Although not produced during the Cold War, this poster
captures the USSR’s deep history of distrust of capitalism.
Furthermore, consider the strategic actions available to the USA and USSR during the Cold War.
Both sides were, first and foremost, capable of nuclear warfare. One of the defining
characteristics of the Cold War was the ‘nuclear arms race’, in which the USA and USSR
competed against one another to test and develop nuclear weapons. By the 1950s, both the USA
and USSR had arsenals of weapons capable of destroying the other nearly entirely (Overy,
2010). Figure 2 charts the development of long-range (intercontinental) and short-range
(regional) nuclear weapons by both superpowers during the Cold War period. The rapid
development of nuclear weapons in the USA is further evidenced by Figure 3, a photograph
taken in 1950 in America of protests against the nuclear arms race. The Soviet nuclear threat
was clearly evidenced in the Cuban Missile Crisis of 1962, during which the USA discovered a
build-up of USSR nuclear missiles in Cuba. A newspaper clip, published the day after the crisis, is
shown in Figure 4. The Cuban Missile Crisis is further described and explained in the Prologue
Magazine article (“Forty years ago”, 2002).
Figure 2: Bar chart of the number of nuclear weapons developed by the USA and USSR between
1955-1980. By the 1950s, each country had sufficiently powerful weapons to obliterate the
other.
Figure 3: Photo of individuals protesting against the use of tax money to develop nuclear
weapons in the USA, 15 March 1950.
Figure 4: Newspaper article from the ‘Daily Sketch’, a British national newspaper, published on
the day following the Cuban Missile Crisis. The article reports on the US President John F.
Kennedy’s infamous TV and radio broadcast, in which he announced the very real possibility of
a devastating nuclear conflict.
Apart from the use of nuclear weapons, each country also had the option of engaging in a
conventional direct conflict with the opposing side, using the same conventional military forces
they had used in World War II (Encyclopaedia Britannica, 2020). A third strategy available to
both countries was to engage in forms of indirect conflict, such as proxy wars. Indeed, during the
Cold War, the USA and USSR entered numerous overseas conflicts, including the Korean War,
Vietnam War and Iran-Iraq war (Blakemore, 2019), in which they armed, equipped or trained
the opposing sides, each aiming to extend their spheres of influence (Overy, 2010). Sargent
(2015) remarks, “There was no systemic war […] Instead, the Cold War’s violence was borne by
[…] people who lived far from the initial geopolitical fault lines”.
Based on this evidence, therefore, we assume that in each stage of the game, the USA and USSR
have three possible strategies: launch a nuclear attack against the opponent (N); engage in a
direct conflict, using conventional military technologies (C); or engage in an indirect conflict, i.e.
support one side of a proxy war in another country (P).
Moreover, we assume the game is structured as an infinitely repeated game. This assumes that,
in each stage, both countries perceive that there is a non-zero probability of the game
continuing to the next stage. This assumption is reasonable because neither country could
anticipate with full certainty when the Cold War would end, as discussed by Hopf and Gaddis
(1993). A ‘stage’ in the game cannot be defined precisely, but is assumed to represent a period
in which both countries observe an opportunity to engage in a nuclear, conventional or indirect
conflict. In each stage, assume the USA and USSR choose their actions ‘simultaneously’; that is,
without knowing the choice made by the opponent. Indeed, this reflects how decisions were
made during the Cold War: strategic decisions made by each country were kept as hidden as
possible (Barnes, 1981).
Finally, consider the payoffs of the game. We make several simplifying assumptions. Firstly,
assume that the USA and USSR have symmetric payoffs and have equal military power. Hence, if
both engage if a non-nuclear war ((C,C) or (P,P)), the outcome will be a tie, and both countries
will face the same military costs (in addition, conventional direct wars are assumed to be more
costly than proxy wars). Moreover, assume countries achieve a higher payoff by engaging in
conventional direct wars rather than proxy wars. Hence, if countries choose (C,P) or (P,C), the
country playing C receives a higher payoff than the country playing P.
Furthermore, assume that, for either country, being attacked with nuclear weapons is the worst
possible scenario. Hence, if Country A launches a nuclear attack against Country B, assume
Country B receives a payoff of -100, regardless of its choice. Moreover, if Country A launches a
nuclear attack and Country B does not, assume that Country A will emerge as the dominant
world superpower in this round, receiving a payoff of 100.
The payoffs for all scenarios, in each stage, are summarised in the matrix below.
USSR
USA N C P
N (-100,-100) (100,-100) (100,-100)
C (-100,100) (-30,-30) (-20,-45)
P (-100,100) (-45,-20) (-15,-15)
Figure 5: Payoff matrix of the stage game.
Predicting the Cold War outcome using the infinitely repeated game
The outcome observed during the Cold War period was that both countries engaged in indirect
conflicts (P) in every period. We show this is a subgame perfect equilibrium (SPE) outcome,
under the following strategy:
• In the first round, play (P,P).
• If N or C was played by either country in any previous round (i.e. (P,P) was not played in
some previous round), play (N,N) in the next round.
• If (P,P) has been played in all previous rounds, play (P,P) in the next round.
The outcome path under this strategy is that both countries play (P,P) forever. We prove that for
some discount factor δ є (0,1), there is no strictly profitable one-shot deviation for either player
under this strategy. As the payoff matrix is symmetric, it is enough to consider one-shot
deviations for the USA only.
Under the strategy, there are two types of subgames:
Type 1: (P,P) was not played in some previous round.
The outcome sequence is (N,N) forever. Each country receives an average payoff of -100.
There are two possible one-shot deviations for the USA. If the USA deviates to (C,N), this results
in the outcome sequence: (C,N) followed by (N,N) forever. The USA still receives an average
payoff of -100. Similarly, if the USA deviates to (P,N), this results in the outcome sequence: (P,N)
followed by (N,N) forever, and the USA still receives an average payoff of -100. Hence, neither of
these one-shot deviations are strictly profitable.
Type 2: It is the first round of the game or (P,P) has been played in all previous rounds.
The outcome sequence is (P,P) forever. Each country receives an average payoff of -15.
There are two possible one-shot deviations for the USA. If the USA deviates to (N,P), this results
in the outcome sequence: (N,P) followed by (N,N) forever. The USA receives an average payoff of
(1 − δ)(100 − 100δ − 100δ! − 100δ” −⋯) = (1 − δ) )100 −
100δ
1 − δ*
= 100(1 − δ) − 100δ
= 100 − 200δ
Therefore, this one-shot deviation is not strictly profitable if and only if
100 − 200δ ≤ −15
⟺ 115 ≤ 200 δ
⟺ δ ≥
115
200 = 0.575
On the other hand, if the USA deviates to (C,P), this results in the outcome sequence: (C,P)
followed by (N,N) forever. The sequence of payoffs for the USA would be -20 in the first round,
followed by -100 forever. As all of these payoffs are less than -15, this one-shot deviation is
never strictly profitable for the USA.
Hence, there are no strictly profitable deviations for any choice of δ such that 0.575 ⩽ δ < 1. For
these values of δ, therefore, the strategy is a subgame perfect equilibrium.
Comparison of the model’s prediction and the Cold War outcome
Similarities
The model’s prediction – the SPE outcome of playing (P,P) forever – shares the same major
characteristics as the Cold War outcome. Both sides avoided engaging in nuclear (N) or
conventional (C) direct conflicts; instead, they sought to extend their political influence by
engaging in indirect conflicts (P) every time they had the opportunity. To support the SPE
outcome, the discount factor in the model, δ, needed to be sufficiently large. This is justifiable as,
during the Cold War, both the USA and USSR desired to maintain their political influence for as
long as possible; hence, both placed adequate weights on their future payoffs when making
strategic decisions, allowing for the SPE outcome to be supported.
The SPE strategy itself also shares similarities with the threats made by both sides during the
Cold War. The notion of ‘mutually assured destruction’ during the Cold War meant that a
nuclear attack from one superpower would be met by an overwhelming nuclear response from
the opposing superpower (Encyclopaedia Britannica, 2020). Moreover, during the Cold War,
even conventional direct conflicts between the USA and USSR would have carried a high risk of
escalating into nuclear conflict. Both these observations are captured by the SPE strategy as it
specifies that, if either superpower plays N or C, the response is to play (N,N) forever.
Differences
A significant difference between the model’s prediction and the observed outcome of the Cold
War is that the model predicts that the game would continue for infinitely many rounds. In
reality, the Cold War did eventually end, due to several factors. Firstly, the USSR suffered a great
economic toll from its defeat in the Afghanistan war (BBC, p.2). Moreover, the USSR faced
growing opposition to Soviet policies in Eastern Europe, culminating in the dismantling of the
Berlin Wall in 1989 (BBC, p.3), and even unrest within the Soviet Union itself (UShistory.org).
Ultimately, the USSR could not afford to maintain the ‘arms race’ against the USA (BBC, p.4), and
in 1991 the Soviet Union was dissolved (UShistory.org).
The key issue with the game theory model, therefore, is that it assumes both countries had the
capacity to continue the game for infinitely many rounds. In reality, after several decades, both
sides had committed a significant amount of resources towards competing in the Cold War. The
opportunity costs of competing, as opposed to accomplishing other national goals, were much
higher than they had been before. Under the infinitely repeated game, however, there was no
option for either country to exit the game, nor was there any limit to how much each country
could commit to competing against each other.
Suggestions for improving the model
The model could be improved by adding the assumption that, in any stage, if a country’s
cumulative discounted payoffs falls below a particular lower limit L, the country prefers to exit
the Cold War, thus ending the game. The limit L represents the maximum total ‘cost’ (in terms of
political, social and economic utility) that the country is willing to incur in its efforts to compete
in the Cold War. In reality, during the Cold War, neither country knew for certain how
committed its opponent was to winning (although their relative economic resources could be
estimated, their relative utilities were unknown). Hence, we can assume that each country’s
limit L is unknown to the opposing country, so that neither country can predict when the game
will end. For this reason, we can assume that both countries would still behave as though they
were playing an infinitely repeated game.
Based on our earlier analysis, therefore, playing (P,P) forever would remain an SPE outcome
when the discount factor is sufficiently high (0.575⩽δ<1). In the SPE outcome path, the USA and
USSR would each receive payoffs of -15 in every round. If they played for infinitely many
rounds, each would receive a sum of discounted payoffs equal to #$%$#&. Hence, if we assume each
country’s lower limit is #$%$#& < 4 < 0, neither country would be willing to play the game for
infinitely many rounds: at some stage, one country’s sum of payoffs would fall below L, and the
country would prefer to exit the game. Moreover, if we assume the lower limits for the USA and
USSR satisfy LUSA < LUSSR, the model would predict that the USSR would exit the game first, as it
would take fewer rounds for the sum of payoffs to exceed LUSSR compared to LUSA. This is
equivalent to assuming the USA was willing to incur a higher cost to compete in the Cold War,
which could reflect the fact that the USA economy was far larger than the Soviet economy
throughout the Cold War period (Directorate of Intelligence, 1985).
Hence, under these additional assumptions, the model would predict that the game would end
after finitely many rounds; the USSR would be the first country willing to exit the game; and as
before, in each round, countries would both choose to fight indirectly. This fits the reality more
closely than the original model’s prediction.
Conclusion
By modelling the Cold War as an infinitely repeated game, we observed that the threat of
escalated nuclear conflict acted as a deterrent for direct conflict between the USA and USSR. The
model predicted that both countries would avoid direct conflict in every round, as observed in
real life. However, it predicted that the game would continue indefinitely, whereas the Cold War
ended after finitely many rounds. We suggested the model could be improved by assuming that
either country could exit the game if it became too costly.
References
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Image references:
Figure 1(a): Catholic Catechetical Guild Educational Society. (1947). Propaganda pamphlet [Illustration].
Retrieved from https://www.nationalgeographic.org/activity/ally-enemy-american-perception-soviet-
union-1920-1950/
Figure 1(b): Deni, V. (1930). Comrade Lenin Cleanses the Earth of Filth [Illustration]. Retrieved from
https://en.wikipedia.org/wiki/Propaganda_in_the_Soviet_Union
Figure 2: Overy, R. (2010). The nuclear balance, 1955-80 [Graph]. In The Times Complete History of the
World (p. 305). London, UK: Times Books.
Figure 3: Palumbo, F. (1950). People picketing the use of tax dollars for the development of nuclear weapons
[Photograph]. Library of Congress, Washington, DC. Retrieved from
https://iowaculture.gov/history/education/educator-resources/primary-source-sets/cold-war
Figure 4: John Frost Newspapers (1962). The Daily Sketch, Tuesday October 23, 1962 [Newspaper clip].
Retrieved from https://www.bl.uk/learning/timeline/item108397.html