# Let’s Play A Game: Choose Effort Optimally (CEO)

Motivation:

Organizations with a hierarchy of people often face the problem of what effort can be expected from the employees given the organizational structure. For example, let us consider the following tree.

Imagine the root node is a Senior Manager of the organization, whom the nodes in the next level (Managers) report to. The leaf nodes are the freshly recruited employees who finished their grad school and joined the organization. The organization has a constant rate ${\lambda}$ of incoming works. All the different employees of the organization, including the Managers and Senior Managers, can exert efforts in order to finish the job. They can choose their working rate ${\lambda_i}$ (effort) to finish the task. However, the nodes who are high above in the hierarchy get a passive benefit due to the work of their subtree. In the figure, the blue nodes in the path shown gets a passive benefit from the work done by the red node. Also, the effort has certain cost to the nodes. So, a node who earns more benefit passively, might optimize his/her payoff by putting a less amount of effort.

The Game:

Let us assume that there are N players connected in some tree structure. Tasks arrive at a Poisson rate ${\lambda}$. Each of the nodes can choose their working rate ${\lambda_i}$, which is also Poisson. Tasks are matched to the agents on a FCFS basis. It can be shown that the expected direct benefit a node gets for choosing effort ${\lambda_i}$ is given by ${\frac{R \cdot \lambda_i}{\sum_{j \in N} \lambda_j}}$. But (s)he also incurs a cost of ${c \cdot \lambda_i}$, c is a constant. In addition, the node gets some passive benefits. Hence the total expected payoff of node ${i}$ is of the form:

Total Payoff ${= \frac{R \cdot \lambda_i}{\sum_{j \in N} \lambda_j} - c \cdot \lambda_i + }$ Passive Benefits.

The passive benefit in this game is shown in the following figure. The nodes in the path to the node get a geometrically diminishing benefit because of the reward got by the red node.

It can be shown that if each player tries to greedily maximize his/her own payoff by changing the effort, this game reaches an equilibrium where nobody can unilaterally change his/her effort and increase individual payoffs.

Imagine You are a Player: