Game Theory The islands of Alpha and Gamma are at war. Alpha has three battleshi
ID: 1097143 • Letter: G
Question
Game Theory
The islands of Alpha and Gamma are at war. Alpha has three battleships and Gamma has four battleships. Battles take place in only two locations: off the coast of Alpha (A) and off the coast of Gamma (G). The naval commander of Alpha may send one, two, or three ships to A and the remainder to G. The naval commander of Gamma may send one, two, three, or four ships to A and the remainder to G. Bigger battles are more important. If Alpha and Gamma send the same number of battleships to location A or to location G, the outcome of the battle is a stand-off (i.e., a tie), with a payout of 0 to both sides. If Alpha sends more ships to a location than does Gamma, Alpha wins that battle and gets a payoff equal to the total number of ships present at that battle, while Gamma loses the battle and gets a payoff equal to the negative of Alpha's payoff. If Gamma sends more ships to a location than does Alpha, Gamma wins that battle and gets a payoff equal to the number of ships present at that battle, while Alpha loses the battle and gets a payoff equal to the negative of Gamma's payoff. The objective of each island is to maximize total payoffs over the two battles.
1. Fill-in the following payoff matrix with total payoffs, where actions describe how many battleships are sent to location A. (Keep in mind that any battleships not sent to A are sent to G.)
Here's the matrix given and I need to find the missing 2 boxes (labeled with ?,?), however I am lost on how to find them. Any help is appreciated!
2
3
GAMMA 1 2 3 4 ALHPA 1 -5,5 ? , ? -1,1 -3,32
-1,1 -3,3 -5,5 -5,53
1,-1 3,-3 -1,1 ? , ?Explanation / Answer
Consider the first blank. Here Gamma sends 2 ships to A and Alpha sends 1. Therefore Gamma wins at A and has a payoff of 3 (the total number of ships present) while Alpha loses and gets -3. The rest of Gamma's (2) and Alpha's (2) ships were sent to G and they draw there. Therefore the total payoff is (-3, 3).
Consider the second blank. Now both of them send all of their ships to A and thus Gamma wins. Gamma gains the total number of ships present, 7 while Alpha gets -7. Hence the payoff is (-7,7).
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