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Saturday, March 29, 2014

### A Crimea Game Tree, with Thoughts on Estonia

Below is a question from the final for my half-semester PhD game theory course. I've put the game tree answer right below. For the other answers, see the comment section, where I'll put them as a comment.

They are the modern analog of Czechoslovakia, as the Crimea is the analog of Austria, Syria the analog of Spain,   the Ukraine proper the analog of Poland, and Finland the analog of the Baltic Republics. NATO is  the analog of France, America the analog of the UK, and  China the analog of the Soviet Union, but fortunately Russia's army isn't analogous to the Werhrmacht and would have to stop with Finland.

Question. Russia’s leaders are either bold or timid. President Obama puts a 70% probability on them being bold. The Russian leaders must decide whether to invade the Crimea or not. If they do, Obama must decide whether to send troops to the Ukraine or not. If he does, then Russia’s leaders must decide whether to withdraw or risk a fight. The status quo payoffs can be set at (0,0)....

...Russia incurs a cost of 100 to send troops to the Crimea, and the US incurs a cost of 100 to send troops to the Ukraine. Permanent possession of the Crimea is worth 400 to Russia, but Russia’s permanent occupation costs America 200. You may assume that if war breaks out, Russia does retain possession of the Crimea. If Russia invades, then it can withdraw from the Crimea later, but it loses 100 in prestige. If Russia invades but does not withdraw, then there is the risk of a war starting between Russia and America, a risk which costs America 500. The risk costs Russia 700 if Russia is timid and 100 if Russia is bold.

(a) (10 points) What is the strategy set of America?
(b) (10 points) What is an example of a weakly dominated strategy for the America?
(c) (20 points) Draw the extensive form, including weakly dominated strategies (there will be 12 end points). You do not need to include the payoffs, though that may be helpful to you. The payoffs correspond to a realistic story, so I suggest in parts (c) and (d) to try to work back from the end without thinking too hard about the specific numbers.
(d) (20 points) What is a pure-strategy equilibrium of this game?

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(a) (10 points) What is the strategy set of America?
Strategy 1 = Send|Invade, Don’t Send|No invasion
Strategy 2 = Send|Invade, Send|No invasion)
Strategy 3 = Don’t Send|Invade, Send|No invaction
Strategy 3 = Don’t Send|Invade, Don’t Send|No invaction

(b) (10 points) What is an example of a weakly dominated strategy for America?