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The Game of Nuclear War

We live in a world with constant threat.  We are worried about terrorist attacks, rapists on the streets, or being in the wrong place at the wrong time.  But, imagine constantly having a nuclear missile pointed at our country.  This threat would cause a great deal of chaos and retaliation.  Israel was faced with this threat due to Iran building missiles.  They had two choices to build missiles as well or not and be defenseless. This experience creates a prisoner’s dilemma.

The players in this simplified game are Israel and Iran.  Yes, in real-life there are many more countries that could effect and would be affected by this game.  But, for the sake of this argument, I am going to focus on Israel and Iran, like the article does.   The two options in this game are to make nuclear missiles and to not make nuclear missiles.  For the payoffs, if both countries create missiles, they will both be at risk of an attack.  They would also both have a defense weapon if they were attacked.  This outcome creates a great deal of fear and most likely one of the countries would be provoked and use their missiles which would result in a war with a great deal of deaths.  Therefore they both have a negative medium payoff.  On a scale of -5 to 5, this payoff would be -3, -3.  If one country created missiles and the other didn’t, there would be a high payoff for the country that made the missiles because they have protection and are not in danger of an attack.  This country has control over the situation.  The other country is in great danger and has a very negative payoff.  They do not have any nuclear warfare to defend themselves if they were under attack.  This payoff would be -5 for the country that does not make the missiles and 5 for the country that does.  The last outcome occurs when neither country decides to partake in nuclear warfare.  This results in a safe environment for both countries.  Even though they do not have a way to defend themselves, there is no nuclear warfare that they need to defend themselves against.  This payoff would be 3, 3.  These payoffs can be seen in the attached chart.

In this prisoner’s dilemma, both countries have a dominant strategy of developing nuclear warfare.  Given that Iran will develop nuclear warfare, Israel receives the best payoff by creating nuclear missiles.  Given that Iran will not develop nuclear warfare, Israel also receives the highest payoff be creating nuclear missiles.  Therefore no matter what Iran does Israel benefits the most by creating nuclear missiles, i.e. their dominant strategy.  This graph is symmetric, so the same logic applies to Iran’s greatest payoff.  This causes the Nash Equilibrium to be both countries choosing to develop nuclear warfare because both countries will play their dominant strategy.

But in terms of social welfare, this outcome does not help the society as a whole.  The social welfare is -6 because both countries now have dangerous weapons and a deadly war will most likely occur.  The social welfare for one country to create nuclear weapons and the other to not is 0.  This is because the society is no better off if one country creates weapons.  The country with nuclear warfare will most likely not use their weapons because the other country will be very afraid and will not provoke them without a defense weapon.  The outcome with the highest social welfare occurs when neither country creates nuclear weapons.  This takes the threat of nuclear warfare out of the picture and saves a great deal of lives.  The social welfare of this situation is 6.  But, it is very hard to get this situation to occur in real life.  This is because both of the countries would have to work together and agree not to be selfish for their own country.  They would have to sign some sort of agreement where both promise to refrain from creating nuclear missiles.  The other problem is, according to this article, Iran has already begun to develop nuclear missiles, so it will be harder to convince them to stop.  So, Israel has three choices, develop nuclear missiles as well, attack Iran’s nuclear plants, or try to get Iran to agree to stop making nuclear missiles peacefully.  As I described above, while peacefully getting Iran to stop making nuclear warfare would be the most beneficial to society, this is very hard to accomplish.  If Israel attacked Iran in order to prevent them from making nuclear missiles, this would be very costly for both countries and would provoke Iran to continue the inevitable, producing nuclear missiles.  Therefore, the easiest thing for Israel to do is to create nuclear missiles as well.  I hope that both countries will be able to see what is best for society and agree to not produce nuclear missiles, but as I have explained in many different ways, this is very unlikely.  So, everyone should expect a nuclear war.

Disclaimer:  In this argument I assumed that nuclear warfare could only be defended by nuclear warfare.  I am unsure as to whether this is true or not, but this assumption helped to simplify my model.  Also, the numbers of the payoffs were assigned based on what I thought the payoff would be on a relative scale from -5, 5.  These are not necessarily the exact payoffs.



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September 2012