This is a good example of sequential games (and with imperfect information). Material is the Gambit sheet Chicken3.gbt and the Excel sheet Chicken3.xls
In October 1962, US surveillance discovered that the Soviet Union was about to install offensive atomic missiles on Cuba. For one week US president Kennedy and his advisors kept this knowledge secret and just discussed among themselves what to do. Robert McNamara had outlined three options: trying to solve the problem politically, surveillance and blockade, or military action against Cuba. When one week later the USA released the information to the public, they went for the blockade option, which they called "quarantine". Russian ships drove towards the blockade, and many thought that the world would slip into an atomic war over this. Eventually, after several days of blockade, some letters between Kennedy and Khrushchev, and some diplomatic talk behind the scenes, the matter got resolved. The Soviets pulled their offensive missiles back from Cuba.
Although the word "game" doesn't seem quite appropriate for a crisis as serious and threatening as this, game theory and in particular the game of CHICKEN is often used to model the Cuba crisis. In the game of CHICKEN both parties move simultaneously, move only once, and have just two options: to stand firm, the "Hawk" move, or to give in, the "Dove" move. If both stand firm, a nuclear war is unavoidable. As far as the payoffs are concerned, at that time, 1962, it had become obvious that a nuclear war would be by far the worst possible outcome for both parties. Both giving in does not change things much, but of course, if one gives in and the other remains firm, the firm party has a slight advantage, both for achieving the goal, and also as a gain of stature in public opinion.
However, this modeling as described could be criticized. Isn't it a little odd that the players have only one move which they perform simultaneously when the conflict lasts 14 days, with several opportunities every day for both parties to give in. Maybe this and other conflicts should rather be described as a multiple step game, where, if you stand firm, you have to confirm your decision repeatedly. In all these rounds the probability of disaster---atomic war---would increase, until eventually such a war would be unavoidable if none of the parties changed their mind. Why would we introduce Nature's (i.e. random) moves in a game like this? Isn't the decision to start the war a decision that can only be made by either Kennedy or Khrushchev? In my opinion the conflict teaches us that this is not the case. There were certain events in the crisis, for instance Soviet boats approaching the blockade line, or a US U-2 spy airplane accidentally entering the Soviet airspace, or an US U-2 spy airplane shot down over Cuba by a missile (ordered by a soviet commander, not by Moskow), that all could have triggered war. One wrong reaction of the soldiers involved, and the presidents wouldn't have had a choice left.
Here is the conflict modeled as an ordinary chicken game. If both play "Dove", if the Russians pull back their missiles from Cuba and the Americans lift the blockade simultaneously, then nothing much happened, expressed by payoffs of 0 for both. If both play "Hawk", then an atomic war occurs, with payoffs of -10 for both---it probably wouldn't even matter who the "winner" of the war is. In the mixed cases the "Hawk" player has some advantage, expressed in a payoff of 1 versus a slight loss, a payoff of -1, for the player playing "Dove". Therefore the bimatrix looks as follows:
| Dove | Hawk | |
| Dove | 0,0 | -1,1 |
| Hawk | 1,-1 | -10,-10 |
There are two pure Nash equilibria, the combinations of Dove versus Hawk. Furthermore there is a mixed Nash equilibrium with 10% Hawk for both, giving a payoff of -0.1 for both (and war in 10%·10% = 1% of the cases). This mixed strategies Nash equilibrium can be computed using the Excel sheet Nash.xls for 2×2 and 3×3 Normal forms; go to the sheet "Nash22" and paste in the bimatrix above.
It should be noted, however, that we have to assume that the payoffs lie on an interval scale if we bring in mixed strategies. In the other two sections, where randomness and expected payoffs occur, we would have to assume an interval scale even for Nash equilibria in pure strategies. This, of course, is a major assumption. Assume someone shows you a dashboard with 1,000,000 buttons. He convinces you that one, and only one of these buttons would trigger a huge and powerful atomic bomb hidden somewhere in your country. He promises $1,000,000 if you press one of these buttons. Of course this is an immoral offer, but if you think the choice is meaningful, you may be close to accepting interval scale.
Next we model the conflict as a multi-step sequential game with imperfect information and randomness, as shown in the figure. Imperfect information occurs since we have simultaneous moves within each step, The reason why we use simultaneous moves in each round, or equivalently sequential moves and imperfect information---no information about the other player's previous move in some cases---will become clear in the exercises maybe, but it can also be explained by the lack of reliable communication channels between the leaders of the Soviet Union and the USA in these days. It took hours until a decision reached the opponent. We assume that if both players play "hawk" in the first round, then there is already a 25% probability of war after the first round.
We will take this opportunity to analyze the game twice. First using generalized backward induction and the Extensive Form in 2.1, and also by finding all pure reduced strategies and the Normal Form in 2.2.
This game has the ordinary chicken game as subgame, the part starting with Adam's second information set, which is just a single vertex. Therefore we use the generalization of backward induction discussed on the subgames page to attack the problem. Therefore we analyze this subgame first. Then we put in the values obtained from any Nash equilibria we find for the subgame as payoffs into Adam's second vertex to find the values for Nature's moves, and then the Nash equilibria for the whole game.
First we concentrate on assigning payoffs to Adam's second move. Of course, arriving at that situation means that both players stood firm in the first round, and they had luck---nature didn't create war yet. Since at this point we just face ordinary one-round chicken, the three Nash equilibria of that situation mentioned above transform into three cases at Adam's second move vertex. So either we have payoffs of -1,1 there, or 1,-1, or -1/10,-1/10. The next step is to compute the expected payoffs at Nature's vertex. We take the weighted average of the -10,-10 payoffs in case of war and the payoffs at Adam's second move, in all three cases. We get payoffs of
Dove versus Hawk in round 2
| Hawk versus Dove in round 2
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Therefore we get a total of nine subgame-perfect Nash equilibira:
Besides the four pure strategies 1), 2), 4), and 5), there are five Nash equilibria with behavior strategies:
Note that (B4) and (B5) look a little funny. They describe in detail what would happen in the second round, although this second round will actually not occur.
Interestingly, in equilibria (B1)-(B3), the payoffs for both players are smaller than those in the mixed Nash equilibrium in the one-round model. Moreover the probabilities for war are higher than there. Thus there seems to be no advantage for the world when moving to the two steps model! Maybe a conflict should rather be short and very dangerous than long with increasing danger?
To practice, let us try to analyze the game again, this time using the normal form of the game: Both players have two information sets, the options "D" (Dove) and "H" (Hawk) in each of them, therefore both have four pure strategies: "DD", "DH", "HD", and "HH". However, if a player plays Dove in the first round, then it doesn't matter what she would have played in the second round, therefore the two strategies "DD" and "DH" are indistinguishable and are therefore subsumed under the reduced strategy "D•".
| D• | HD | HH | |
| D• | 0, 0 | -1, 1 | -1, 1 |
| HD | 1, -1 | -5/2, -5/2 | -13/4, -7/4 |
| HH | 1, -1 | -7/4, -13/4 | -10, -10 |
When we solve this normal form game, using the Excel sheet Nash.xls again we get four pure Nash equilibria: D• versus HD, D• versus HH, and also HD versus D• and HH versus D•. There are also three mixed Nash equilibria, and two infinite families of half-mixed Nash equilibria:
Obviously the four Nash equilibria in pure strategies are just the extreme cases of the families (M4,a) and (M5,a)---they are (M4,0), (M4,1), (M5,0), and (M5,1). All these Nash equilibria of the form (M4,a) and (M5,a) are weak in the sense that a deviation from the Nash equilibrium by one player (the one playing the mix of HD and HH) does not necessarily reduce the payoff for that player, but may keep it constant.
The solutions computed in sections 2.1 and 2.2 should be identical. This is obvious for the pure strategies, but how do the five behavior strategies (B1)-(B5) relate to the three mixed strategies (M1)-(M3)?
Let's start translating (B1)-(B5) into mixed strategies, using the formula on the page about Behavior Strategies. Then (B1) translates into both players choosing D• with probability 63/103, HD with probability (40/103)·(9/10) = 36/103, and HH with probability (40/103)·(1/10) = 4/103. This is obviously mixed strategy (M1). In the same way, the pairs of behavior strategies (B2) and (B3) translate into the pairs of mixed strategies (M2) and (M3). (B4) translates into Ann playing 9/10 of HD and 1/10 of HH, and Beth playing pure strategy D•. This is (M4,9/10). In the same way, behavior strategy (B5) translates onto mixed strategy (M5,9/10).
Let us now conversely translate the mixed strategies into the behavior strategies. Remember that at each player's first vertex the probability for "Hawk" should be P(HD)+P(HH), and the probability for "Dove" there should be P(D•). At the second vertex, the probability for "Hawk" should be P(HH)/(P(HD)+P(HH)), and the probability for "Dove" should be P(HD)/(P(HD)+P(HH)). If we translate all five mixed strategies, we get the following:
The last two cases, (C1) and (C2) are only subgame-prefect if they coincide with (B4) and (B5).
Discussion of whether mixed strategies do make sense here. This is probably not a repeated game, or is it? Would Kennedy and Khrushchev sit down and roll dice? Maybe they should, in order to remain unpredictable!
We want to model that the probability of war increases as time goes by, therefore we choose 1/4 and 1/2 as probabilities for war after the first and the second round. If the third round is reached, and both parties still stand firm, war is certain. The figure displays the extensive form of the corresponding three-round version.
Since every player has three information sets with two choices at each of them, each player has 2·2·2=8 pure strategies. We abbreviate them by three letter words with the letters "H" and "D", where the first letter refers to the choice in the first round, and so on.
Again some of these pure strategies can be identified, since they always lead to identical outcomes. For instance, the four strategies "DDD", "DDH", "DHD", and "DHH" can be subsumed under the reduced strategy "D••". If a player chickens in the first round, the other two rounds will not evolve. In the same way the two pure strategies "HDD" and "HDH" are combined under a new name "HD•". Therefore each player has the four reduced strategies "D••", "HD•", "HHD", and "HHH". The payoff matrix looks as follows:
| D•• | HD• | HHD | HHH | |
| D•• | 0, 0 | -1, 1 | -1, 1 | -1, 1 |
| HD• | 1, -1 | -10/4, -10/4 | -13/4, -7/4 | -13/4, -7/4 |
| HHD | 1,-1 | -7/4, -13/4 | -50/8, -50/8 | -53/8, -47/8 |
| HHH | 1,-1 | -7/4, -13/4 | -47/8, -53/8 | -10, -10 |
The Nash equilibria can be classifies as follows: