Part 1: Dominant and dominated strategies In the following games, identify any dominant or dominated strategies. Specify whether the strategy is strictly or weakly dominant/dominated. Then use dominant and/or dominated strategies to provide the solution to the game. Player 2 Player 1 Left Right Up 4, 1 2, 4 Down 1, 2 3, 3
For the questions that ask you to draw the games, you can either draw them on the computer, or draw them by hand and then insert a picture. The table function in word works well for drawing these games, and was what I used.
Part 1: Dominant and dominated strategies
In the following games, identify any dominant or dominated strategies. Specify whether the strategy is strictly or weakly dominant/dominated. Then use dominant and/or dominated strategies to provide the solution to the game.
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Player 2
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Player 1 |
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Left |
Right |
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Up |
4, 1 |
2, 4 |
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Down |
1, 2 |
3, 3 |
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Player 2 |
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Player 1 |
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Left |
Center |
Right |
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Up |
7, 3 |
8, 6 |
4, 9 |
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Middle |
1, 8 |
5, 7 |
3, 4 |
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Down |
2, 1 |
9, 5 |
6, 2 |
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Player 2 |
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Player 1 |
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Left |
Center |
Right |
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Up |
2, 1 |
3, 6 |
7, 5 |
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Middle |
0, 8 |
7, 7 |
3, 5 |
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Down |
2, 2 |
4, 1 |
3, 1 |
- Assume there is some deadly airborne disease. Assume that wearing a mask does nothing to protect oneself from getting the disease, but would limit how much the wearer would spread the disease if they got infected. Two people are considering whether to wear a mask. Because wearing a mask would protect the other person, they each get 50 utility for the other person wearing a mask. One of the two people is benevolent, and gets an addition 30 utility from wearing a mask themself, as they do not want to pass the virus on to the other person. The other person is selfish and finds masks somewhat uncomfortable, and therefore gets -10 utility for wearing a mask. (The utilities compound, so you have to add them together as appropriate). Draw the game. Identify any dominant or dominated strategies. Then use these to solve the game. Does this game tell us anything about the politics of wearing masks to respond to COVID-19? (Note that elements of the assumed scenario do not exactly match our understanding of the effect of masks on COVID-19, and it does appear that masks provide some protection to the wearer as well). If there is one element of the game that you would modify to better represent the politics of wearing masks to respond to COVID-19, what would it be?
Part 2: Pure Strategy Nash Equilibria
In the following games, find all pure strategy Nash equilibria
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Player 2
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Player 1 |
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Left |
Right |
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Up |
0, 0 |
3, 4 |
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Down |
2, 1 |
3, 2 |
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Player 2
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Player 1 |
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Left |
Center |
Right |
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Up |
2, 2 |
0, 0 |
2, 2 |
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Down |
2, 2 |
5, 5 |
0, 0 |
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Player 2 |
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Player 1 |
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Left |
Center |
Right |
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Up |
2, 2 |
3, 2 |
0, 3 |
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Middle |
3, 1 |
1, 3 |
2, 3 |
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Down |
8, 0 |
3, 4 |
2, 2 |
- A Democratic Senator in West Dakota is retiring. The incumbent representative (to the House) is considering running for that seat, as well as a political newcomer. Both are Democrats. Thus, both the incumbent and the newcomer have three options: They can run for Senate, they can run for the House seat, or they can stay out of both races. Each candidate prefers the Senate seat to the House seat. However, the incumbent believes that he is next in line to the Senate seat, so gets 150 utility for winning the Senate seat, and only 50 utility for winning the House seat. While the newcomer would prefer the Senate seat, she would be relatively happy to get either, so gets 100 utility for the Senate and 80 utility for the House seat. Both candidates get 0 utility for no seat. The state is heavily Democratic, and so if only one candidate runs for a seat, they are nearly guaranteed to win that seat. If both candidates run, they have a 50% chance of winning the primary, after which they would be nearly certain to win the election. (Think expected utility for how to calculate their preference). Write down the game, then find all pure strategy Nash equilibria. Does this game tell us anything about when candidates might choose to run for office?
Part 3: Mixed Strategy Nash Equilibria
Find all Nash equilibria, including both pure and mixed strategy equilibria
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Player 2
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Player 1 |
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Left |
Right |
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Up |
4, 0 |
0, 6 |
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Down |
0, 2 |
8, 0 |
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Player 2
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Player 1 |
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Left |
Right |
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Up |
4, 2 |
2, 3 |
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Down |
2, 8 |
6, 3 |
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Player 2
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Player 1 |
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Left |
Right |
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Up |
0, 0 |
7, 3 |
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Down |
4, 4 |
2, 0 |
- Recently, a major question in the news is why the Afghan army collapsed so quickly and unexpectedly in the face of the Taliban offensive. To help understand the situation, consider the following simplified model of the situation. Assume that there are two Afghan soldiers in the same unit. Each can stay and fight the Taliban or desert. If both stay and fight, they will win any battle against the Taliban, which each values at 100 utility. If either soldier deserts, the Taliban wins, which they each values at 0 utility. In addition, if only one soldier stays, that soldier will be killed by the Taliban, which he values at -1000 utility. In addition, the Taliban is offering a bounty to any soldier that deserts. One soldier does not care about this, and would not take the bounty, while the other values the bounty at 50 utility. Draw the game based on this scenario. Find all pure strategy and mixed strategy equilibria to the game. What does this game and the equilibria say about why the Afghan army collapses so suddenly?
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