Game Theory, Nash Equilibrium, and Grinders

There is a sandwich shop on campus that sells open-faced toasted subs. At this restaurant there are four managers who participate in a kind of iterated prisoners dilemma, though I'll try to show how each kind of game theory exercise we worked in class can build a model of this scenario.

More about the situation: Since, no manager started working in the store when it opened for business, each of these four players joined a game already in progress. They work in a regular round robin rotation, that is, manager 1 works the first shift; manager 2 works the second shift, and so on with manager 1 coming back in to work the fifth shift, etc. Due to a hilarious love square between these managers none of them communicate outside of work or during shift changes. Finally, each manager really needs to keep her job so they will do whatever a customer asks; they are fully willing to do the the amount of work required to serve all the customers and this alone will prevent termination.

Let's assume Monday morning manager 1 (M1) walks into the store and starts to get things ready. M1 finds that there are some things that are not done that should be done. Now, M1 has a decision to make, either she can work feverishly to get all the things done that need to be done or she can choose to do the least amount of work required to serve the customers and no more. That is, she can serve all the guests but refuse to restock or prepare food for potential need in the next shift. She decides to choose the latter. M2 works that night and finds many things undone and has the choice laid before M1 that morning. Suppose, she chooses the latter on the basis of a tit-for-tat strategy, and suppose M1 did as well. M3, same outcome. When M4 comes into the shift she chooses the second option again but on the basis that she has been burned once, and from now on will never cooperate (Nuclear option).

This complicated real world case highlights a couple of things. One kind game theoretic model for this game is that each manager knows that her dominate strategy for "gettin' paid" and doing the least amount of work is to do only what is required by the customer. This creates a Nash Equilibrium. It is in the best interest of each manager to do the least amount of work possible and none are given an incentive to change.

Another interesting perspective of this example is the following. In lab, when we played the group games with the common pot, it only took one person to remain a freeloader in the initial round, where everyone was naive enough to think everyone would put all their money in the pot, to cause others to defect as well (the round where the number of defectors was identified). One person's decision had a major impact on the rest of the players. That's the kind of impact that M4 has on the rest of the group. If we were to add one new manager each week with the tit-for-tat strategy but who initially chooses to cooperate, M4 will be able to change any consecutive choices to cooperate into a new trend to defect.

The working world is even more complicated than this example since there are consequences to not doing anything during a shift because a manager can complain to someone further up the authority chain, which is about the most obvious of options in a pool of extra complications. The models that we work on in class are extremely telling even while making a bunch of extra assumptions. Think about how much more money we each could have made in lab if we were able to talk to all the members of the group. On second thought, there's extra credit for being in some earnings percentile, I think people would have still chosen to defect since in the long run we are in competition. Interesting stuff!