# Strike time in a game

Suppose A forwards a puzzle to B where if B solves the puzzle within 24 hours then it gets a payoff of M (M>0), if not then it gets nothing i.e. 0. Consider A locks the money with the bank for the 24hours and if B solves the puzzle it withdraws the money from the bank. If not bank refunds M to A after 24 hours. If B solves the puzzle within 10seconds (strike time 10), it gets payoff of M. If it solves at 23rd hour (strike time 23rd hr) it gets a payoff of M. Can there be someway to disincentivize delay in solving the puzzle so that A need not keep his money locked for long period of time? One strategy which I think is that ask B to lock a certain amount as well, so that it is forced to solve the puzzle quickly. How do I calculate the strike time at which B solves so that any delay beyond this will not be profitable?

## 1 Answer

Calculating a precise strike time without additional details is - in my opinion quite clearly - impossible. On the question of how to do this with additional info/assumptions:

A straightforward solution would be to make M time dependent - the larger the time elapsed, the smaller M is. Sounds like you do not want to do this.

Base problem

Let us now assume M is not time dependent, and B maximizes the present value of her payoff (assume a constant interest rate $$r$$) minus the cost of the effort of completing the task at time $$t$$ (denoted by $$C_t$$), so the goal of B is

$$\max_t \frac{\text{M}}{(1+r)^t} - C_t.$$

$$C_t$$ not dependent on time

If we also assume that $$C_t$$ is constant in time, then this is not only decreasing in $$t$$, but decreasing at a decreasing rate. (The second derivative w.r.t. $$t$$ is positive.) This means that B will either not want to lose any present value and solve the puzzle at the earliest possible time (when $$r$$ is quite large), or that B will not care about the loss very much and solve at the deadline (when $$r$$ is quite small).

B also puts money down

This is unchanged by B also escrowing (locking up) an amount of money ($$\text{M}_B$$), as the loss $$\text{M}_B - \frac{\text{M}_B}{(1+r)^t}$$ resulting from this is also decreasing at a decreasing rate.

$$C_t$$ dependent on time

Let us now assume that $$C_t$$ is time dependent, e.g. B has to be at the movies for the next three hours and can only solve the puzzle afterwards, or the computational costs make it costly to lease additional computing power and cut down on time. If there are instances where $$C_t$$ increases at an increasing rate, that is $$C_{t+2} - C_{t+1} > C_{t+1} - C_{t}$$, then it is possible that the maximization problem will have an interior solution, i.e. the optimal time will be larger than zero but smaller than the deadline. However, one can only calculate this if one knows the interest rate $$r$$ as well as the cost function of B, so one needs to get quite a bit of information from B.

$$r$$ dependent on time

Oddly if we allow $$r$$ to vary, that is the interest rate is $$r_t > 0$$ at time $$t$$, not much changes. In the $$C_t$$ not dependent on time case, the problem will again be solved immediately or just at the deadline. If $$C_t$$ is dependent on time, an interior solution is possible, but to calculate the optimal time we need a bunch of additional information/assumptions.

• Thanks @Giskard for the explanation. On what factors must I make the rate of interest dependent? For my case, M is a cryptocurrency say Bitcoin. Against what should I measure the rise/fall in interest rate r? Also is there any condition like B has to lock a greater penalty than the reward M i.e. $M_B> M$? Or if thats the case, B might not be interested to solve the puzzle? – Subhra Mazumdar May 7 at 4:22
• Hi! 1. "Against what should I measure the rise/fall in interest rate r?" I don't know. This depends on the goals and assets of B. 2. "Also is there any condition like B has to lock a greater penalty than the reward M i.e. M_B> M? Or if thats the case, B might not be interested to solve the puzzle?" I don't really understand what you mean here; but in my answer I wrote that B escrowing some money does not really change anything. – Giskard May 7 at 4:27
• Sorry but I didn't get what is meant by assets of B or is it based on how B perceives the value of M depreciates with time or not? From the equation you have stated $max_t \frac{M}{(1+r)^t}-C_t$, for my case $C_t$ is constant, then this makes t=0 as the trivial solution. I need the value of $M_B$ so that the reward $B$ gets by delaying must be greater than the loss it incurs by keeping $M_B$ locked. If there exists a time $t'<24 \ hrs$, such that the reward becomes less than the loss then $B$ will not delay. Hence I was asking whether $M_B$ must be set higher or lower than $M$ – Subhra Mazumdar May 7 at 4:32
• Sorry, but it seems like you have not yet fully understood my answer, or I don't understand what you are trying to do. (Why set $M_B > 0$ at all?) – Giskard May 7 at 4:37
• But how the second derivative of $\frac{1}{(1+r)^t}$ with respect to $t$ is negative? – Subhra Mazumdar May 7 at 4:55