I am looking for help in designing a multi-level game for a network simulator that I build.
In this network, the actors have the choice of setting up a number of units and filling their units with money. They have a fixed amount of money to start with, and strive to get the most financial reward possible.
Before the game starts, each actor has a heterogeneous amount of money and no units.
A pot of money is distributed to the players according to their relative size in the market, which results from the number of units and the amount of money in each unit. In particular, each unit receives a score, and the proportion of the reward that can be claimed for that unit is determined by the score of that unit divided by the total score of all units in the network. Logically, an actor's reward is the sum of his unit's scores divided by the scores of all units in the network.
The score of a unit is divided into two parts: operation and investment. One unit receives 0.7 points for operation (it is assumed that all units reach at least this score), and one unit can receive up to 0.3 points for the amount of investment in the node.
There is a minimum investment of $ 20 for a unit to qualify for rewards. If a unit has the minimum investment, it will receive 50% of the total investment potential – 0.15. This gives a node with the minimum investment amount a score of 0.85.
For all above-minimum investments, there are falling marginal returns. The optimal investment is $ 50, which gives 90% of the total possible investment points – 0.27 – and a total score of 0.97
As a further reference, the percentages of the total possible investment points (0.3) are given for each investment step.
$ 20 – 50% (from 0.3) -> .15
25 – 65%
30 – 75%
$ 35 – 81%
$ 40 – 85%
$ 45 – 88%
$ 50 – 90%
This is calculated by an Arctan decreasing marginal yield function whose second derivative is 0 at $ 50.
How can I write an equation that shows how an actor chooses, how many units to build and how much money to put in each unit?