Arms Race, Defense Budget Allocation and Prisoner’s Dilemma


In an earlier post we discussed the classic Prisoner’s Dilemma problem where rational unilateral (or individualistic) decisions fare worse than co-operative actions. A common real world example is the increasing defense budget, observed across the world. Defense Budgets are generally allocated as a fixed % of GDP. Countries spend on an average 2-4% of their GDP on military preparedness and defense. What is the optimum spend and can this be reduced and reallocated more efficiently towards other growth promoting government actions?


As a tax payer, I am often fascinated by the state’s allocation of the Budget. I belong to the school of thought where Budget should be allocated efficiently keeping the principles of Cost-Benefit in mind. Benefits should be improved Social welfare, equity, Health Status, Education level, employment, GDP growth, Political influence (indirectly for Strategic advantage culminating into other Direct Benefits) etc.

I pulled out Defense Spend and GDP for 9 countries and ranked them based on their absolute spend (in US Billion$). Before we look at the actual data, let us form some intuition on what we should expect.
a) Poorer countries (Yemen, Eritrea) would spend less on Military and more on socio-economic growth drivers like Infrastructure, Health, Education, Housing etc.
b) Developing countries with a history of war (India, Pakistan), international and domestic conflict, would spend more on Defense.
c) Developed smaller countries (Luxembourg, Singapore) with very high per capita income, would spend less on Defense and more on social security, Health and unemployment benefits.
d) Due to recent acts of terrorism and their participation in resolving international conflicts, larger Developed Nations (US, UK) would allocate more on Defense. Military spend to flaunt Political power and influence.

Now let us look at the data,
Rank Country SIPRI Military spend in USD $B (2009) IMF GDP in USD $B (2009) Military spend as % of GDP
1 United States $ 663.26 $ 14,119.05 4.7%
3 United Kingdom $ 69.27 $ 2,178.86 3.2%
10 India $ 36.60 $ 1,235.98 3.0%
25 Singapore $ 7.97 $ 182.23 4.4%
35 Pakistan $ 4.82 $ 161.99 3.0%
40 Malaysia $ 4.08 $ 192.96 2.1%
68 Yemen $ 1.20 $ 25.13 4.8%
87 Luxembourg $ 0.41 $ 52.43 0.8%
94 Eritrea $ 0.33 $ 1.87 17.5%

In most cases, our observations are quite consistent with our intuition. US has above average spend on Defense, which has been a topic of serious debate in recent times. Trends suggest, a drop in oversees bases in Iraq and Afghanistan, and more spend on Job Creation and Health.

Eritrea, an impoverished country in the horn of Africa is an interesting case study. It has been in constant conflict with it’s neighboring nations of Yemen, Sudan and Ethiopia. It is an Authoritarian state, with a population of over 5Million with a GDP just around 1.9Billion.  Yemen on the other hand is a presidential republic and started economic reforms in late 90s with aids from IMF and international donors. Yemen also has a significant Geopolitical influence with it’s strategic location and oil reserves.

Singapore, stood out, in my view as well. Singapore spends 4.4% on Military, a percentage which has stood somewhat constant over the years. The Government deems it necessary to maintain the sovereignty of this small city state. Singapore fears sudden political developments in the region, particularly in the neighboring states of Malaysia and Indonesia, which might require its armed forces to be used either as a deterrent, or as a means of national defense. The fact that it takes a very long time to build up such capabilities has meant that Singapore's military development has been continuous and sustained. This development also reflects other factors, such as political will, aided by the longevity of the same regime in power since independence; Singapore's sustained economic development which has given it the ability to devote resources to defense.

India and Pakistan have had 4 major wars (‘47,’65,’71,’99) along with frequent border conflicts. Both had a similar % spend (3%) of GDP for military spend, which as a % of GDP is at par with Global average.
A thing to note is, that since Defense spends are in most cases incurred by Government, this when translated into a % of their annual budget could be a really high figure. For instance, US Government spent 19% of their budget last year in Defense.

Economists worldwide agree that ~1.5 Trillion USD (2009 estimate) spent collective by all nations on military spend could have been put to better use, had their been more effective multilateral agreements on peace and cooperation. The Prisoner’s Dilemma is applicable here for precisely the same reason. Since countries don’t trust each other and often question the UN’s ability to mediate conflicts, decisions on military spend are thus unilateral. Each country puts up the most it can spend on defense. Had they all co-operated, they needed just enough to resolve internal domestic conflicts, leaving the rest to a better use.

Also, if in a particular year, India decides to raise its Military budget to say 3.1% of its GDP, this will trigger panic among its neighbors. It could be both Positive (against fear of new threats) and Negative (Geo-Political influence, war, territorial conflict). Pakistan and China would raise their spend just to signal their willingness to counter India’s move.

Thus the pareto-optimal solution would have been no one spends anything on international/cross-border defense, had countries cooperated and had there been an effective way to enforce it. However the NASH equilibrium we always end up is each country spending the maximum they can afford to spend.

http://en.wikipedia.org/wiki/List_of_countries_by_military_expenditures
http://en.wikipedia.org/wiki/Yemen
http://en.wikipedia.org/wiki/Eritrea
http://www.visualeconomics.com/military-spending-worldwide/

Prisoner’s Dilemma: Where Equilibrium is not always Socially Optimum

interrogatorPrisoner’s Dilemma is a very popular situation enacted innumerable times in modern cinema and television shows. The situation is very simple.

  Two suspects Shawn and Gus are arrested by the police. The police have insufficient evidence  for a conviction, and, having separated both prisoners, visit each of them to offer the same deal: if one testifies (defects) for the prosecution against the other and the other remains silent, the betrayer goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge.

If each betrays the other, each receives a five-year sentence. Each prisoner must make the choice of whether to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How should the prisoners act?


A prisoner’s action depends a lot on how the other convict reacts to the same situation. Game theory can be used to predict the outcome. However, we will see, Prisoner’s Dilemma is a bit unique.

Sentence for (Shawn, Gus) Gus Defects Gus remains silent
Shawn Defects 5 years, 5 years 0 years, 10 years
Shawn remains silent 10 years, 0 years 6 months, 6months

From the payout matrix, if they were great friends and trusted each other, they know that the other wouldn’t defect. Hence they could play a co-operative strategy with both remaining silent and be sentenced for 6 months each. This is the Pareto Optimal solution where both benefit.

A Pareto optimal solution is one where no “one” person can be made better off without making someone else worse off than before. The Pareto Optimal solution is that both remain silent (Silent, Silent). Their combined sentence is 1 year. For Shawn to be made better off (go free), he has to defect making Gus worse off than before.

However, if both of them were rational and not confident of the other’s reaction, they would try and maximize their individual payoffs. Thus they will always defect since, irrespective of the other person’s decision, defecting always results in a comparatively better payoff (5 years vs. 10 years or 0 vs. 6 months). Defect, Defect is the dominant strategy for both.

The NASH equilibrium is this self-interest maximizing solution, which is both of them confessing and ending up with a sentence of 5 years each. The beauty of this non-iterated Prisoner’s dilemma problem is that this equilibrium is not Pareto optimal as before.

The economic moral of the puzzle is that a group whose members pursue rational self-interest may all end up worse off than a group whose members act contrary to rational self-interest. More generally, if the payoffs are not assumed to represent self-interest, a group whose members rationally pursue any goals may all meet less success than if they had not rationally pursued their goals individually.

Simply put, in most situations at least one would always confess (driven by self interest) and the cops eventually have someone to frame and they solve the case :)

Now, look around you for instances of Prisoner’s Dilemma. You would be surprised how common it is. But then, that’s for the next post.

For the math lovers,

If Shawn and Gus were two rational mathematicians, they would gauge the relative probability of the other person defecting and decide accordingly based on the total payoff. And the probability would be based on how much they can trust each other. However, sadly, in this particular instance (in Prisoner’s Dilemma problem), defecting is the dominant move irrespective of the probabilities.

It can be easily shown that Payoff for Shawn if he defects is always higher.
Sentence for Shawn if Shawn is Silent > Sentence for Shawn if Shawn defects
[10x + 0.5(1-x)] > 5x + 0 * (1 – x)
4.5x + 0.5 > 0
For all +ve value of x

where x is the probability between 0 and 1, of Gus defecting
Thus irrespective of Gus’ move, Shawn will always defect.