How To Survive A Group Project Using Game Theory
Game theory is a set of mathematical tools to analyse strategic behaviours by rational players, whose actions are contingent on the strategy implemented by one another, while their actions influence the result of the game.
Let’s suppose you and Alex are working on a group project. Ideally, both of you should put in the work but you realised you are better off playing video games while letting Alex do everything. However, you will take the risk of Alex being an unreliable groupmate who too refuses to work, where failing the project is something you do not want.
The above situation can be modelled by constructing a game theory table. The numbers in the grid represent ‘utility,’ an abstract concept in economics that refers to the degree of satisfaction an individual receives from an act.
If you believe Alex is going to work, you should not work to obtain a higher utility (70 is greater than 50). If you believe Alex is not going to work, you are better off not working (30 is greater than 25). This implies no matter what Alex does, you should not work. Since ‘not working’ leads to a strictly higher payoff than action ‘work,’ we would say action ‘not work’ strictly dominates action ‘work’. A rational player will never take a strictly dominated action.
Now consider the profile of actions from Alex’s perspective. Using similar logic as above or using the fact that this is a symmetric game (payoffs are symmetrical), it can be deduced that Alex would also not work, regardless of your actions. We have solved the game! We reached an equilibrium where both of you do not work, with a payoff of 30 each. Such an equilibrium is defined when players do not achieve a higher payoff deviating from their initially chosen strategy, assuming the other players also keep their strategies unchanged.
An interesting situation arises. If you both work, the 50, 50 payoff is clearly more desirable.
However, game theory tells us, that without collaboration, not working is the only rational choice. Suppose you and Alex are working on infinitely many projects, one after another. If you both collaborate and choose ‘work’ every time, you will be receiving a continuous payoff stream of 50. What if you betray? You will gain a payoff of 70 once but Alex will not be too happy about it. Alex would stop collaborating with you, and as revenge pick ‘not work’ for the rest of the projects, ensuring that you get the lowest payoffs possible (you can at best get 30 compared to 50 or 70). What Alex has implemented is called a grim trigger strategy, where a player remains cooperative until the opponent defects, in which the player defects for the remainder of the game.
After some algebra, the two payoffs can be equated to obtain r=1 or δ=0.5. Under such a discount factor, you should be indifferent between collaborating and betraying. For δ> 0.5, future projects are not discounted too heavily so you will still benefit from the stream of 50 from collaborating as opposed to just 30. On the other hand, for δ<0.5, there is a high weighting on the initial projects so you should betray to maximise your payoffs.
By all means, blame game theory for not getting involved in the project and good luck explaining it to your former friend Alex.
Edited and reviewed by Tannish Bagga.
References/Wider Reading
https://www.investopedia.com/ask/answers/032615/what-concept-utility-microeconomics.asp
Ronny Razin, “EC1A3 Microeconomics I” (class lecture, 2021)
https://corporatefinanceinstitute.com/resources/knowledge/economics/nash-equilibrium-game-theory/