Suppose two players 1 and 2 are working on a joint project. When completed, the

Question

Suppose two players 1 and 2 are working on a joint project. When completed, the project is worth 16 to each player if completed in periods 1 or 2, 6 if completed in periods
3 or 4 and nothing if not completed by period 4. The project takes 4 units of effort to complete.Players move sequentially with Player 1 moving first. Each can choose to put in 0,1,2 or 3 units of effort with the cost of x units of effort being x squared: All effort is observable to everyone.
Each players payoff is the value of the project if completed minus the total cost of effort expended by him or her. Find a subgame perfect equilibrium

Explanation / Answer

Nash equilibrium is a point in game theory when all players in the game are playing best possible strategies and have no incentive to change their strategy to get a better payoff. A subgame perfect equilibrium or subgame perfect nash equilibrium is when there is a nash equlbrium in the subset of a game.

Now, it is given to us that there are two players 1 and 2 working on a project that is worth 16 each if completed in period 1 and 2, it is worth 6 if competed in period 3 or 4 and it is worth 0 if completed after period 4. It is also given to us that both players can choose effort of either 0,1,2 or 3. The cost is square of the effort, that is effort2 and the payoff is value of the project - (effort)2 for each player.

Let us assume that both players will prefer that they complete the project in periods 1 or 2 as it gives them the highest value of 16. The payoff to each player in this case will be value - (effort)2. We can write this as :

Payoff to players = 16 - e2 where e is the effort put in by the players.

Let us assume that player 1 starts in period 1 by putting no effort. In this case, even if player 2 puts the maximum effort of 3 units the project will not be completed as it requires four units of effort and both players will have to move to period 2.

The project has to be completed in this period for them to get the highest value of 16.

Now, lets assume player 1 decides to put effort 1 and player 2 puts effort 3. In this case, the payoffs will be

Payoff for player 1 = 16 - 12 = 16 -1 = 15

Payoff for player 2 = 16 - 32 = 16 - 9 = 7

However, this is not a nash equilbrium as it will not be accpetable by player 2 and he/she will not put in the required effort of 3. This will cause the project to move to period 3. However, player 1 and 2 both do not wish this as it gives them a lower value of 6.

Anticipating this, player one will put in effort 2 in period 2 and player 2 will put in effort of 2 also. This will give them payoffs of :

Payoff to player 1 = 16 - 22 = 16 - 4 = 12

Payoff to player 2 = 16 - 22 = 16 - 4 = 12

Both players put equal effort of 2 and get equal payoffs of 12. So, this is the subgame perfect equilbrium in the given game when the project is completed in period 2 and both players put in effort 2 and a get a payoff of 12 each.

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