Background People of the modern world are divided against the existence of nuclear warheads. Therefore, the game that will be analyzed is the nuclear summit scene from G.I. Joe: Retaliation by John M. Chu. In order to understand this scene it is necessary to know that Zartan, a soldier of the Cobra clan, disguised himself as the President of the United States. The scene consists of leaders of Israel, United Kingdom, France, India, North Korea, Russia, and Zartan. The scene begins by talking about reduction of nuclear warheads by 50% by 2016, however, Zartan wishes to have total disarmament as of right now. When the other leaders revolt by saying that the world is safer with nuclear weapons Zartan presses the button to deploy all the nuclear warheads that the United States have, hence stage 1 of the game begins. Once the warheads are in air the leaders of the other countries become flustered and one by one start to deploy their own nuclear war heads towards each other. Once all players deployed the nuclear missiles, Zartan asks if anyone is willing to abort the madness of bombing every country in the world. When no one decides to be the first to abort it’s own nuclear war heads, Zartan self-destructs all of the United States’ nuclear missiles in order to pursue his original goal of a total nuclear free world, hence stage 2 starts. Once Zartan destroyed all the missiles in mid-air, he begins to tell the other leaders to abort and self-destruct their missiles unless they want to destroy the world. As none of the leaders wants to take the blame of creating a mass genocide, all the leaders destroy their own nuclear missiles creating a nuclear free world. As Zartan reaches the goal of total nuclear disarmament, the Cobra Leader enters the scene and introduces the advanced weapon technology created by the Cobra clan. When the asymmetric information between the United States and the other countries are revealed, the leaders of the other countries become outraged. In order to stop the Cobra Clan from destroying their countries, the Cobra Leader wants total submissiveness to the Cobra Clan from each leader leaving all the leaders in a hard spot. It is clear that Zartan initiated all his actions expecting to achieve the ultimate goal of nuclear free world in the end. The team will explore the strategic game theory that show how Zartan was so sure to win due to the setup and the payoffs of the game.

Assumption Although there are 7 players including the US, we will use the US as a trigger that starts the game and group the other 6 countries into two: European/Middle (EU) and Asian countries for simplicity. As professor have mentioned, if the our team were to draw out the first sequential game with 7 players and 2 actions each, the outcome would come out to 128 moves. The gametree with the two groups will be sufficient enough to show the payoffs and the psychological interpretation of all the countries. Also, the outcome of the grouped countries will be shown in simple integers. We understand that human lives and millions of military budget cannot be compared to simple numbers, however, the calculation of each stage will provide sufficient data to validate the perfect evil plan that Zartan has. All of the payoff in both stage 1 and 2 will be written in this order (EU, A).
Structure
Because U.S. already knew that they had the “secret weapon” and want all others to get rid of their nuclear bombs after they launched them, the choice that U.S. is going to make is already determined, which is to Fire at stage 1, and Cancel at stage 2. This will be the initial move that will begin each of the stages. Two players, EU and Asia, are playing the games without having all the information because only U.S. knew the information that they had a stronger secret weapon other than nuclear bombs.
Figure 1. Stage 1 Structure
US FIRE

Sequential Game Scenario (Stage 1)
In the first stage, we looked at the game as a sequential game. In this summit meeting, U.S. is the one who first launched the nuclear bomb, and E.U. is the next person who decides what to do due to the fact that he or she sits right next to the president of United States. EU, who is the first mover in this case, has the option to either fire back or do nothing. The next move is then passed over to Asia countries, who must decide whether they are going to counterattack or do nothing after E.U.’s decision. Even if Asian countries think that European Union would counter attack, they still have 2 options; the option of firing and do nothing. The game tree above graphically describes the sequential game scenario of this so-called Nuclear War. All the strategies possible and the rollback equilibrium strategy are as follows:
Figure 2. Possible Strategies for each country All possible strategies Rollback Equilibrium
EU F1, Do Nothing F1
Asia F2F3, F2N3, N2F3, N2N3 F2F3

This would result in a payoff of (-1,-1) which means that everyone fires their nuclear missiles. This brings us to the stage 2 when US suddenly disarms all of its nuclear weapons in the air.
Simultaneous Game Scenario (Stage 2)
Figure 3. Stage 2 Structure US Cancel

Asia
European Union Cancel Keep
Cancel 0,0 -2,-1
Keep -1,-2 -2,-2

Now, let’s approach the next stage from the simultaneous game perspective. In the second stage, after all the nuclear bombs are launched, US decide to cancel the bombing while it’s up in the air. This is already determined just like at stage 1, since U.S. is the only one who already knew the top-secret information. Our goal here is to find the nash equilibrium and in what randomness each player will play the game. While all the nuclear bombs are in the air, either Europeans or Asians have choice to stay put or cancel the bombing, knowing that U.S. already put out their bombing. The possibility of mixed strategies in this game structure will be able to represent the psychological affect each other’s probability of choosing either cancel or keep will affect the decision of the other country group.
Payoffs
The payoffs in these games will be shown in simple positive and negative integers. Stage 1 shows a sequential game in which 2 countries can decide to either fire their nuclear weapons or not, given that the US already fired the weapon. As you can see in figure 1, all of the outcomes in stage 1 is negative due to all the destructions and lives lost as a result of a nuclear war. If a country is to fire and get hit, the outcome would be -1 since the country at least got to fight back as a strong nation. On the other hand, the group that only gets attacked but not fight back would have the outcome of -2 since the submissiveness of the government could be portrayed as a weak nation to the citizens. Stage 2 is little bit different because the movie scene shows that as soon as the US decides to disarm the missiles in the air, all the countries have to decide simultaneously if they should cancel the missile as well. The simultaneousness is incorporated in this stage because the missiles will reach the continents soon and all the countries are under time pressure. The payoffs in this table represent the grouped countries’ image to the public. If both of the groups cancel the attack, the public will not find out about the war in general so highest payoff in this chart is 0 for both of the grouped countries. When the nuclear missile is launched, even if one group cancel the missiles, due to the action of the other group, the world will now figure out that there was such happening. The fact that the group who canceled the missile still launched the missile in the first place will leave them with the public image of -1. The group who kept its attack order will receive the image of -2 because they will be destroying so many of the countries. Keeping the missile attack from either of the groups will always lead to negative outcome because the secret Nuclear summit and the information about stage 1 will be revealed no matter what. The worst case is the (-2,-2) which is if both of the countries keep its’ attacks and the world finds out about it.
Solution
Sequential Game
To analyze a sequential game, we first constructed a game tree mapping out all of the possibilities then we followed the basic strategic rule. Looking at the game tree map, let us look ahead and reason back. First of all, the very last decision and assume that if it comes to the point where Asian countries have to make decision on rather to fire or not fire the nuclear warhead when European or Middle East countries decided to fire. The deciding player, Asian countries, will choose his or her optimal outcome that is the highest payoff or otherwise most desirable result of which in this case is to fire with the outcome of -1. Second, when assuming that the European or Middle East countries decided to do nothing with the existing nuclear warheads, Asian countries should fire since -1 is greater than -2. Now, let us back up to the first deciding player, European or Middle Countries assuming the next player would choose his or her best outcome. From here, we will treat the following decision as fixed because we have already decided what that second player will pick. When continuing this process of reasoning back in this way and having Asian countries’ best decision in mind, the final decision is that the European or Middle East countries should choose to fire because from (-1,1) and (-2,-1), -1 is better outcome to the European countries. Therefore, the rollback equilibrium of this sequential game between the European or Middle East countries and Asian countries is (-1,1).
Notice that this procedure assumes that the players are as smart as each other, and are doing the same analysis. While this may not be the case, it is the only safe assumption. If it is correct, we will have made our best possible decision. For it to be incorrect, an opponent must choose an option that is not in his or her own best interests. Also, notice that looking ahead and reasoning back determines not just one player’s optimal strategy, but those for all players. Once the solution has been determined, which in this case is for the European or Middle East countries to fire and the following Asian countries to fire as well, it is irrelevant whether or not the game is actually played, as no one can possibly do better than the solution dictates. Alternatively, one could argue that the player who gets to make the last decision wins. As this situation is determined as a sequential games, ultimately, there are only two choices: either the player with the last decision gets his or her best outcome, or the game is not played. Thus, the game tree obviates the need to actually play out the game.
Simultaneous Game
Both European/Middle East and Asian countries have decided to fire their nuclear weapons. Now, the United States cancel their nuclear warheads in the air. Unlike in the first sequential game, both European/Middle East and Asian countries play a simultaneous game without knowing the other player’s actions. In order to find the Nash Equilibrium, it is necessary to find the best outcome for the both players where they would not want to deviate their actions.
In the simultaneous game, figure 3, suppose EU chooses to cancel their nuclear weapons, Asian countries have two choice of cancelling and keeping theirs which gives the outcome of 0 and -2 respectively. Asian countries will also cancel since it gives higher output of 0. In case of EU choose to maintain their nuclear weapons, Asian countries are indifferent to any choices for both choices yield same output of -2.
On the other hand, looking at the game in perspective of European/Middle East countries, suppose Asian countries cancel, EU countries will choose to cancel for the outcome of 0 over -1 when they keep the weapons. However, if Asian countries do not cancel, EU countries end up with the same output of -2 whether they cancel the weapons or not. Thus, there are two Nash Equilibria, (Cancel, Cancel) and (Keep, Keep).
In this mixed strategy, assume the EU countries’ probability to cancel is (p) and (1-p) to keep the weapons, while the Asian countries’ probability to cancel is (q) and (1-q) to keep.

As shown in the calculations above, probabilities for both countries to cancel nuclear warheads are zero. According to the probabilities we have now, neither of players will disarm the missiles. However, if there is any possibility of larger than zero, both of the players will cancel the missiles since it gives higher outcome of (0,0) than (-2, -2) from keeping.

Name of Country Number of Nuclear Bomb
USA 7,315
India 90-110
Russia 8,000
France 300
North Korea