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Suppose we have a hungry fox. He has a gigantic bunch of spoiled carrots that he cannot eat (and wouldn't eat if they were fresh anyway), but he knows the local bunnies in the neighboring area love carrots.

He seals up all the spoiled carrots in an air-tight basket and heads over to put a mischievous plan into effect. He announces to all the $n$ number of bunnies in the region that he is raffling off a huge trove of carrots worth some $x$ amount of money that he could get at a neighboring market, but which he is deciding to raffle off to his good neighbors. He says he will deliver the heavy load himself to the winner's house privately.

The fox insists on keeping the carrots fresh and keeps the basket closed, so the bunnies cannot smell or see the carrots before the raffle, at the moment. He plans to sell tickets for $p$ money each, and then randomly pick one of the tickets as the winner. A bunny can buy more than one ticket. This process is public and verifiable. When he delivers the spoiled carrots, he will act surprised and then refund any tickets that the winner had bought, and tell the winner he will give refunds to the rest of the bunnies too...before fleeing with the rest of the proceeds before anyone can stop him.

The bunnies are somewhat suspicious of this whole raffle, to varying degrees, but if they were the winner, they would not ask for a replacement prize of equal value to the carrots and would accept the refundwhile the fox was at their house (it's a cultural thing). We say each bunnies' expected utility for tickets is:

$$\mathbb{E}[u_i(t_i, g_i)] = \frac{t_i}{\sum_i^n t_i}(g_i\cdot[C - x^2 + x] - pt_i) + (1 - \frac{t_i}{\sum_i^n t_i})(-pt_i)$$

where $C>0$ is some constant, and $g_i \in (0, 1]$ is a uniform gullibility distribution (higher is more gullible, each bunny has a different $g_i$ somewhere in the distirbution). Notice how if the fox announces $x$ as too high, the bunnies will think the raffle is too good to be true and start valuing the raffle less than they would have, for all else equal. $\mathbb{E}(u_i)$ is public information, and the distribution of $g_i$ is public info as well.


My question is whether or not it is possible for the fox to

  • determine the demand for tickets given the information
  • if so, what $x$ and $p$ he should announce to maximize expected profit
  • if not, what additional information the question needs to be a compelling, solvable problem
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  • $\begingroup$ If I understand the question correctly, this is a two stage game. The second stage is a $n$-player all pay auction, whose solution, a function of $x,p$, should be easily obtainable. Then given the second stage equilibrium outcome, the fox maximizes expected revenue by choosing $x,p$ in the first stage. $\endgroup$ – Herr K. Dec 19 '15 at 19:09
  • $\begingroup$ @HerrK. It's not quite an all pay auction, since the winner is not whoever spends the most money on tickets, but whoever's ticket gets picked out of all the tickets that were purchased. $\endgroup$ – Kitsune Cavalry Dec 19 '15 at 19:19
  • $\begingroup$ Is it correct that each $g_i$ is independently distributed? If so, then you can try to solve for a symmetric Bayesian Nash equilibrium in the second stage game, which would give the demand for tickets. Let $s(g_i)$ be the symmetric equilibrium strategy as a function of type $g_i$. Then, a typical bunny with $g_i$ would have strategy $$s(g_i)=\arg\max_{t_i} \int_0^1 \frac{t_i}{t_i+(n-1)s(g_j)}g_i(C-x^2+x)dg_j-pt_i.$$ $\endgroup$ – Herr K. Dec 20 '15 at 2:30
  • $\begingroup$ Yeah, $g_i$ is independently distributed. Are you building off of Alecos' answer or is it more of a game theoretic approach? $\endgroup$ – Kitsune Cavalry Dec 20 '15 at 2:47
  • $\begingroup$ I think as long as the bunnies are strategic (i.e. each tries to maximize its own payoff, taking into account other bunnies' decisions and $x,p$), then it's only natural to take a game theoretic approach. Alecos' answer assumes bunnies are identical, but in your formulation, the bunnies are only ex ante identical. Their decisions on $t_i$ is a function of their realized gullibility $g_i$. That's why I thought it would make sense to consider a symmetric BNE in the stage where bunnies choose, and, in anticipation of such an equilibrium, the fox decides on $x,p$ in the first stage. $\endgroup$ – Herr K. Dec 20 '15 at 11:39
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A critical point here is to note that the total number of tickets is not set a priori. This is good, because it makes the expected utility function non-linear in $t_i$, and so permits us to proceed (half-way).

Writing $S$ for the total number of tickets and $S_{-i}$ for the total number minus the purchases of bunny $i$, and simplifying, the expected utility is

$$\mathbb{E}[u_i(t_i, g_i)] = \frac{t_i}{S}\cdot g_i\cdot [C-x^2+x] -pt_i \tag{1}$$

The first order condition for utility maximization of one bunny with respect to number of tickets bought is,

$$\frac {\partial \mathbb{E}[u_i(t_i, g_i)]}{\partial t_i} = \frac{S_{-i}}{S^2}g_i\cdot [C-x^2+x] - p=0$$

$$\implies t_i = \left(\frac {S_{-i} g_i\cdot [C-x^2+x]}{p}\right)^{1/2} - S_{-i} \tag{2}$$

The second-order condition is satisfied so this will be a maximum. Rearranging $(2)$ we obtain

$$S = \left(\frac {S_{-i} g_i\cdot [C-x^2+x]}{p}\right)^{1/2} \tag{3}$$

The choice of $i$ was arbitrary so we have

$$S_{-i} g_i = S_{-j} g_j,\;\;\; \forall i\neq j \implies (S-t_i)g_i = (S-t_j)g_j$$

$$\implies t_j = S - \frac {g_i}{g_j}(S-t_i), \;\;\; \forall j\neq i \tag{4}$$

Summing over $j\neq i$ we obtain

$$S-t_i =S_{-i} = (n-1)S - (S-t_i)g_i\sum_{j\neq i}g_j^{-1} $$

$$\implies (S-t_i) = \frac {n-1}{1+g_i\sum_{j\neq i}g_j^{-1}}S \tag{5}$$

Inserting $(5)$ into $(3)$ we get

$$S = \left(\frac {\frac {n-1}{1+g_i\sum_{j\neq i}g_j^{-1}}S g_i\cdot [C-x^2+x]}{p}\right)^{1/2}$$

$$\implies S = \frac {(n-1)g_i\cdot [C-x^2+x]}{\left(1+g_i\sum_{j\neq i}g_j^{-1}\right)p} \tag{6}$$

We were able to express total demand as a function of the decision variables of the fox, and the parameters/random variables of the model. Nevertheless it also hints at the problem here (multiply by $p$ to get the revenue function), but let's derive it explicitly.

For later use, from $(5)$ we also get

$$t_i = S\left(1-\frac {n-1}{1+g_i\sum_{j\neq i}g_j^{-1}}\right) \tag{7}$$

Turning to the profit function of the fox, it has certain gross revenues equal to $pS$ and then she will have to payback the amount paid by the bunny that gets to win the lottery. So with probability $t_i/S$ the fox gets $pS - pt_i$. So the expected profit function, after ticket sales have been finalized and before the lottery draw, is

$$E(\pi) = \sum_{i=1}^n \frac {t_i}{S}\left(pS - pt_i\right) = p\sum_{i=1}^n \frac {t_i(S-t_i)}{S} \tag{8}$$

Inserting $(5),(7)$ into $(8)$, we have

$$E(\pi) = p\sum_{i=1}^n \frac {S\left(1-\frac {n-1}{1+g_i\sum_{j\neq i}g_j^{-1}}\right)\left(\frac {n-1}{1+g_i\sum_{j\neq i}g_j^{-1}}S\right)}{S} $$

$$= pS\sum_{i=1}^n \left(1-\frac {n-1}{1+g_i\sum_{j\neq i}g_j^{-1}}\right)\left(\frac {n-1}{1+g_i\sum_{j\neq i}g_j^{-1}}\right)$$

Using also $(6)$ we get, after simplification

$$E(\pi) = \frac {(n-1)^2g_i\cdot [C-x^2+x]}{1+g_i\sum_{j\neq i}g_j^{-1}}\cdot \sum_{i=1}^n \left[\frac {\left(g_i\sum_{j\neq i}g_j^{-1}-n+2\right)}{\left(1+g_i\sum_{j\neq i}g_j^{-1}\right)^2}\right] \tag{9}$$

Equation $(9)$ reveals the issues here: While from $(3)$ the fox can infer all actually realized $g_i$'s, ex ante, the expected profit function looks like a very complicated function of $n$ Uniform $(0,1)$ random variables.

But most importantly, profit does not depend on price (since to begin with Total Revenue does not depend on price). While this is standard in a perfectly competitive environment (where market equilibrium determines price), here we have a monopoly. To fix this, one should go back to the expected utility function and change its quasi-linear form, and assume instead concave utility in $pt_i$, $v(pt_i), v'>0, v''<0$. This will maintain price as an argument of the profit function together with $x$, and maximization of profit with respect to $(p,x)$ jointly could be attempted.

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