I am looking for some help in designing a multi-level game for a network simulator I am building.
In this network, actors have a choice to set up a number of units and to fill their units with money. They have a fixed amount of money that they start out with and are aiming to gain the earn the largest monetary reward they can.
Before the game starts, each actor has a heterogeneous amount of money and no units.
A pot of money will be distributed to the actors according to their relative size in the marketplace, which is determined by the number of units they have and the amount of money in each unit. Specifically, each unit will receive a score and the proportion of the reward claimable for that unit is determined by that unit's score divided by the total of all units' scores in the network. Logically, an actor's reward is the sum of their units' scores divided by all the units' scores in the network.
A unit's score is divided into two parts: operations and investment. A unit receives 0.7 points for operating (it is assumed all units will gain at least this score), and a unit can receive up to 0.3 points for the level of investment in the node.
There is a minimum investment of $20 for a unit to be eligible for rewards. When a unit has the minimum investment, it earns 50% of the total possible investment points - 0.15. This gives a node with the minimum investment amount a score of .85.
For all investment above the minimum, there are diminishing marginal returns with the optimal investment being $50, which gives 90% of the total possible investment points - 0.27 - and an overall score of .97
For further reference, here are the percentages of the total possible investment points (0.3) for each investment level.
$20 - 50% (of 0.3) --> .15
$25 - 65%
$30 - 75%
$35 - 81%
$40 - 85%
$45 - 88%
$50 - 90%
This is calculated by an arctan diminishing marginal returns function, the second derivative of which 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?