# Form of security with a potentially infinite amount of equity for a continous payout for game?

I'm trying to create a game that's a bit like monopoly. The objective of the game is to earn money by investing money. The game has a limited amount of players and an increasing amount of money each game. That is done through regular stocks with a fixed supply but also through a system that involves buying shares of a revenue stream whose intensity changes through the course of the game.
For the game, its very important that users can buy into that stream at any given time with any given amount of money. It should not be possible to limit access to that revenue stream by simply buying up all shares and then refusing to sell them at any price.
I don't want to get into to much detail about how the game would work, its just important that this revenue stream remains open to all players at any given time for any lump sum.
But that does not mean its always a good idea to invest in it, but it should still be profitable for the players to correctly predict incoming revenue.

The "shares" can be bought directly from the "bank" and traded amongst players (but it would not be a problem if buying directly from the bank would be a better option in every case).
A share in the revenue stream would grant the player a portion of it. The revenue stream should always be divided amongst players as the money comes in (it should never flow to the bank). Also shares cannot be sold back to the bank.

Now since the "bank" would be an autmated system and not an active player in the game, it needs a system to automatically determine the price of the shares. And this is where I am stuck. How can I incentivize players to buy into a portion of something that is reduced over time? One idea that I had was to set the price of shares equal to the amount of shares. That way the shares should always arrive at the ideal equilibrium and people getting in early are rewarded. The problem with this is, that if the revenue stream is expected to go down, almost nobody will buy new shares, essentially cutting them off from buying it.
So can somebody help me create a system like this? A system that

• rewards getting early
• is always open to new investors
• reduces equity as more people buy in
• Hi! I am into games and economic systems, but I have to ask (other members of the SE): is this on-topic? Apr 29 '21 at 14:15
• I was pretty sure it isnt on topic on money st. So I posted it here Apr 29 '21 at 14:46
• I am pretty sure it is not on-topic on History either, but that does not mean you should post it on Physics. You need not worry about this, question is for the other members, discussion will probably move to Meta. Apr 29 '21 at 15:39

One "natural" approach would be to base the price on the expected discounted revenue stream. Writing $$R_t$$ as the (random) revenue in period $$t$$ of one share, this then would be: $$p_t = \mathbb{E}\left(\sum_{t = 0}^T \frac{R_t}{(1+r)^t}\right) = \sum_{t = 0}^T \frac{\mathbb{E}(R_t)}{(1+r)^t}.$$ In addition, you might want to lower this price a bit, especially if the income stream is very volatile and if you would expect that individuals are risk averse.
One thing determine then is the rate $$r$$. Ideally, this should be set equal to the rate of return of a risk-free asset (if you have this in your game). If the game does not last very long, I guess setting $$r = 0$$ also makes sense.