Chicken
You’re driving along a straight road when all of a sudden, you see a car at the other end, headed straight towards you in a collision course. You have to act quickly; do you swerve, or do you head straight forward, hoping the other car will swerve?
Or, consider this situation. You and another person have to split a certain resource, such as food. You can either try to cooperate with them and split the food evenly, or you can be a bit more competitive and try to take more of the food for yourself. The other person has the same options. However, if you both try to compete for the food, you’ll end up fighting and wasting more energy than the food is worth.
Crash!
Let’s create a payoff matrix to represent the situation involving the collision course. If both players swerve, then both people gain or lose nothing, and an accident is avoided. We can give this outcome a value of 0 for each player. If one person swerves while the other goes straight, then the person who swerves suffers a loss of pride while the person who goes straight gains pride. We can give the player who swerves a value of -17 and the player who goes straight a value of +17, since the loss and gain from either person is relatively small. If neither person swerves, then they both crash, which is an extreme loss. We can give this outcome a value of -6511 for each player. The matrix that results is as follows:
The Nash equilibria occur when the two players act differently, with one swerving and the other going straight. There is also a mixed strategy Nash equilibrium, where each player swerves or goes straight at random, and with certain probabilities. Finding the mixed strategy is left as an exercise to the reader.
Biology
A very similar game occurs in evolutionary biology, between two people competing for food. This is the classic Hawk-Dove situation, where a dove is generally defined as a person who tries to cooperate, and a hawk is generally defined as a person who tries to compete. Let’s create a similar payoff matrix to represent this situation. Let’s assume we have a food with a value of 34. If two doves interact, they split the food evenly, so each dove gains a value of +17. If a hawk and a dove interact, then the hawk takes the food for themself and leaves the dove with nothing. We assign the hawk a value of 34 and the dove a value of 0. And if two hawks interact, they fight over the food, possibly injuring each other and wasting so much energy that the payoff is negative. We assign each hawk a value of -17. Here is the matrix representation of this situation:
Again, the Nash equilibria occur when the two players act differently, with one playing dove and the other playing hawk. Once again, there’s a mixed strategy Nash equilibrium, and finding it is left as an exercise to the reader (I love writing that). It’s also interesting to note that if the payout for Hawk/Hawk or Straight/Straight was greater than the payout for playing dove while the other person plays hawk or swerving while the other player goes straight, then the scenarios would become a Prisoner’s Dilemma, where both players would be incentivized to play hawk or go straight.
Conclusion
The existence of the mixed strategy Nash equilibrium demonstrates that sometimes, it’s not ideal to constantly play dove or hawk, and a balance between the two is optimal. And I think this reveals something that isn’t taken into account when looking at these simple matrices for one-off interactions: players can build trust. In problems such as the Infinite Prisoner’s Dilemma, strategies that tend to cooperate, tend to have a higher payoff. I don’t know if the same situation occurs with the Infinite Chicken Game or the Infinite Hawk-Dove Problem, but if you can build trust with other players, then maybe, that is encouraging enough to play dove just a bit more.
Life Update!
The first quarter is over. I fortunately maintained an A in AP Gov despite not studying for the final like I promised myself I would. I was pretty stressed about the final but I got an A in all my classes, which I guess is nice. My classes this second quarter will be much easier though, so I have a lot more time to finish up all the college applications I’ve been procrastinating on. It feels like I’ve had less time as of lately, however. I know it’s only temporary, as college applications are not something I will have to worry about after December, but it still sucks, as I have less time to do what I enjoy.
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