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We show that a group of individuals who coordinate their betting has a strategy to obtain positive expected gains in a “fair” lottery. To illustrate, consider a $1,000$ number lottery in which a “crowd” of one thousand individuals each purchase a \$1 quick pick (sampling with replacement) and a coordinating “syndicate” that acts as a single bettor and purchases one of each ticket combination for a total of \$1,000. On average the crowd bets nothing on $36.8\%$ of the tickets, has one combination on $36.8\%$ and two or more on the remaining ticket combinations. Since the syndicate always has exactly one winning ticket, its expected payoff can be calculated as a function of the number of the tickets held by the crowd. When the crowd has no winning ticket, the syndicate wins the entire jackpot of \$2,000. When the crowd has one winning ticket, the syndicate wins \$1,000 = \$2,000/2, and so on. Thus the syndicate's expected payoff is
where the first term is the contribution to the expected payoff when the crowd has no winning ticket, and the second when the crowd has exactly one winning ticket. Therefore the syndicate's expected return is positive even without terms involving the crowd's holding of 2 or more winners!
This paper has three parts. First, the expected returns are calculated for a simple equiprobable $1,000$ number lottery that has no take.\footnote{The take is the fractional amount a lottery deducts from the betting pool.} We show that a syndicate that bets $1,000$ different tickets earns on average a $26.41\%$ return against a crowd of $1,000$ small players who independently bet one ticket each. A syndicate strategy of buying $n < 1,000$ different tickets is not profitable unless $n \ge 583$.
Then, results are proven for equiprobable lotteries with $t$ tickets and no take. If the crowd bets $\$c$ and a syndicate bets $\$s$, it is demonstrated that the syndicate has a positive expected return if $s > (1 + cy)/(1 - y)$, where $y = (1 - t^{-1})^{c + 1}$ and it bets its tickets on a set which is as evenly distributed as possible. If $s \le t$ and $c \ge 2$, it is shown that the Nash equilibrium for the syndicate and crowd occurs when the syndicate chooses $s = t$ and the crowd chooses quick picks from an uniform distribution. It is also shown that small coordinating groups in the crowd have little impact on the syndicate's returns.
Finally, the equiprobable condition is dropped, there is a take on the betting pool and there is a carryover pool. We prove that (1) the syndicate can always achieve a better-than-fair split of the jackpot pool, (2) the best asymptotic strategies for the syndicate and crowd consist of betting aligned with ticket probabilities (probability-proportional betting), (3) the syndicate's expected return is greatest when the lottery is equiprobable, and (4) if a crowd is risk averse or risk seeking, its asymptotic expected return is lower than it would be with a probability-proportional strategy.
The fact that the syndicate has a winning strategy is due to the basic logic of coalition formation in games (myerson1997game). Each player $i \in N$ in a noncooperative game has a “reserve” value $\nu_i$ that consists of the minimum payoff the player can achieve when acting alone. But to each coalition $S$ of players, there will be a total payoff achieveable through cooperation, $\nu(S)$, which is never less than the sum of the reservation values $\nu_i$ of coalition members, $\sum_{i \in S} \nu_i$. We show that a lottery game in which players in $S$ choose to act as a coalition and ones in $N \, \backslash \, S$ choose to act independently produces an excess expected value for the coalition: $\nu(S) > \sum_{i \in S} \nu_i$. There is no surprise here --- it is quite plausible that coalitions will have edges.
An early contribution to betting strategy in government-sponsored lotteries is chernoff1980analysis. As a statistician, Chernoff knew about “digit preference” in, for example, reported ages in censuses -- there are too many reported ages ending in '$0$' and '$5$'. Using the Massachusetts State Lottery having numbers 0001 to 9999, Chernoff found that numbers ending in '$0$' and '$9$' were unpopular, and he surmised that tickets combining many unpopular numbers might be underrepresented in lottery betting pools. If this were the case, then betting only greatly underrepresented numbers would constitute a winning betting system. Following up on this idea, ziemba1986dr report many other numbers that are unpopular in the Canadian “6/49” Lottery using weekly data of marginal number frequencies published by the British Colombia and Western Canada Lottery Corporations. Among the unpopular numbers in the Canadian 6/49 lotto were 1, 10, 20, 28, 29, 30, 32, 34 and all but two numbers over 38, giving a total of 19 statistically unpopular numbers. The authors cite popularity of birthdate months and days and the geometry of the ticket layout for the majority right-handed players as reasons for persistent popularity and unpopularity, and recognize that some numbers can be temporarily overbet when they penetrate the public's awareness. They found that unpopular numbers were stable over time, but in later years, Ziemba's finding that the unpopularities had changed somewhat and had regressed toward the mean provided evidence of a modest learning effect (Ziemba, personal communication). Using empirical probabilities available for the 6/49 numbers and assuming approximate independence of number probabilities and no carryover pool, ziemba1986dr estimate the expected return per dollar wagered for a generic ticket $(i_1, i_2, \ldots, i_6)$ as
where the factor $\$0.45$ adjusts for the lottery take and consolation pools, and each factor $F_i$ measures the ratio of equal number probabilities (=$1/49$) to $i$'s estimated betting probability. In lotteries with carryover pools, the factor $\$0.45$ is higher.
Using formula (ref), the authors investigated the strategy of betting tickets involving only the 19 unpopular numbers and subsets thereof, and find that those strategies win so rarely as to be unattractive as practical systems. RePEc:inm:ormnsc:v:38:y:1992:i:11:p:1562-1585 investigate an optimal 6/49 Lotto strategy that bets small fractions (to maximize the expected logarithm of final wealth subject to a ruin-avoidance condition) of one's capital using a variant of formula (ref). The authors come to a discouraging conclusion --- that it takes millions of years to achieve a favorable result with high probability!
We discuss next the literature relevant to our paper's formulas. 10.2307/2117538 show a formula for the expected value of a syndicate
where $W$ is the number of tickets bet by the syndicate, $c$ is the cost of one combination, $k$ is the fraction of the handle going into the jackpot, $N$ is number of bets by the crowd, $R$ is the carryover from previous drawings, and $p$ is the probability of winning in a single play. Using the notation of Section (ref), formula (ref) with $c=1$ is essentially a Poisson approximation to formula (ref) multiplied by the jackpot
The authors do not mention that expectation (ref) is positive when expression (ref) $> 0$ and $k=1$, the main assertion of our paper. The purpose of their paper was to advise on economies of scale in lotteries (e.g., multi-state lotteries), not to discuss (ref) as a potential winning strategy. Of the papers surveyed here, 970220123219961201 provide the most complete account of the expected values for lottery strategies, including the purchase of one of each ticket, a strategy the authors call the “trump ticket.” Their formulas are then used to evaluate the “fairness” of lotteries in the context of externalities. No analysis of optimal strategies or recognition of the strategic value of a trump ticket is discussed. RePEc:hcx:wpaper:1109 also discuss the trump ticket. While the authors acknowledge that the return per ticket is generally better when the trump ticket is bet than when a single ticket is bet, there is no discussion of the trump ticket as a potentially winning strategy or its role in a Nash equilibrium.
Much empirical research on lotteries has been done on racetrack parimutuel pools and sports betting. These studies agree that (a) parimutuel odds are consistent with race or game outcomes citeulike:81468 and (b) there is a persistent favorite-longshot bias (FLB) in individual races which results in underbetting of favorites and overbetting of longshots (Ziemba2008183). Several non-mutually exclusive explanations have been offered to explain the FLB bias: (1) poor estimation of probabilities, (2) inside bettors, (3) preference for risk or skewness, (4) heterogeneous beliefs, (5) market power of an uninformed bookmaker, (6) constrained arbitrage and (7) last minute betting by informed players. To this we add a mitigating factor: the presence or absence of syndicates. Many other details of Pick 6's and other lottery-like racetack parlays can be found in Ziemba:Adventures,Ziemba:ExoticBetting.
A sizable segment of the literature discusses the possibility of winning betting systems at the racetrack or in sports, such as RePEc:inm:ormnsc:v:27:y:1981:i:12:p:1435-1452 and citeulike:1337256, but they concentrate mostly on individual races or simple parlays, not on full-blown lotteries.
In this paper we consider pure jackpot lotteries that award only one prize, a jackpot that consists of a carryover pool (possibly 0) from the previous drawing and a fraction of the monies wagered for the current drawing. The jackpot is shared equally among all who hold the winning ticket selected at the drawing. If no one holds the winning ticket, the current jackpot pool becomes the carryover pool for the next drawing.
There are two groups of bettors: a syndicate that coordinates its betting and a crowd that does not; the manner of (in)coordination is described below. We use the following notation:
Since the marginal distribution of each component of a multinomial distribution is binomial, the distribution of the number of winning tickets held by the crowd is
We define a probability $t$-vector $e_s$ by
so that
where $1$ is a $t$-vector of ones.
Thus the syndicate will hold $s_{\!_{D}}$ and the crowd, $K_{\!_{D}}$ winning tickets. The random win $W(s,c,p,r,q,a,x)$, gain $G(s,c,p,r,q,a,x)$ and return $R(s,c,p,r,q,a,x)$ to the syndicate are
where it is understood that a ratio $0/(0 + 0) = 0$ in (ref). The syndicate's expected gain and expected return are
where $v = a + (s + c)(1-x)$ is the jackpot and $s_i/(s_i \, + \, k) = 0$ when $s_i = k = 0$ in (ref). In the following, we use notation such as $G(s)$ or $R(s,r)$ to indicate that we study (ref), (ref) or (ref) as functions of the variables indicated, other variables having specified values.
Suppose you are strolling in the park one fine day and see a lottery stand offering a one-day special. The proprietor informs you that she will return in equal shares to the winners all monies bet --- she will reserve no portion for herself. She explains that there are \(1,000\) tickets each costing \(\$1\), and that \(1,000\) people have already bet a \$1 “quick pick”, a ticket chosen randomly from an uniform distribution with replacement. She says that currently, this $\$1,000$ constitutes the entire prize pool and if no one wins it, it becomes part of tomorrow's jackpot. She says she will close the betting soon and asks if you want to participate. You just happen to have \(\$1,000\) in your pocket. Should you bet? \\
We assume that your goal is to maximize your expected return. At first thought, betting seems unwise because the lottery is “fair” in the sense that everyone has the same opportunity to win. On the other hand, your composite opponent has played a very bad strategy indeed -- that of picking each lottery ticket randomly, independent of previous choices. That scheme of picking tickets leads by chance to some numbers being bet twice, some three times and some not at all.
But that scheme is inferior to always picking unbet tickets. To see this, imagine you act as the crowd's proxy and will bet the \(\$1,000\) for them. Suppose \(s < 1,000\) different tickets have been bet and consider how to bet the $(s+1)$-st, either on (a) a ticket already bet (the \(i^{th}\)), or (b) on one not yet bet (the \(j^{th}\)). Which choice yields the greater expected return? Choices (a) and (b) have the same payoffs on every ticket drawn except for the \(i^{th}\) and \(j^{th}\). Assume that the distribution of tickets the crowd bets on the \(i^{th}\) and \(j^{th}\) are the same, and \(k\) are already bet. If the \(i^{th}\) ticket is drawn, the payoffs to the two strategies are (a) \(2/(k+2)\) and (b) \(1/(k+1)\), while if the \(j^{th}\) is drawn, the payoffs are (a) \(0\) and (b) \(1/(k+1)\). Because these two possibilities are drawn with the same probability, their sums can be compared --- that of (a)'s two cases, \(2/(k+2)\) with (b)'s, \(2/(k+1)\). But \(2/(k+1) > 2/(k+2) \) showing that it is always better to choose an unbet ticket.
This reduces considerably the candidate strategies. Only those which bet $s$ different tickets, \(0 \le s \le 1,000\), are admissible. Since in an equiprobable lottery, every set of $s$ different tickets occurs with the same probability, the particular tickets selected do not affect calculations of expectations. Using the notation from Section (ref) with $a = x = 0$, $t = c = 1000$, $p_i = q_i = 1/t$ and $s_i$ as above, we calculate $R(s)$, the syndicate's return from betting $s$ different tickets.
Why does the crowd lose? The answer: as a composite player, theirs is an inferior strategy. The only ways they can avoid this problem are to subscribe to a coordination mechanism that doesn't duplicate the tickets they bet (cooperate) or to bet nothing (the Nash equilibrium when utility is linear). Such a mechanism could be provided, for example, by a quick-pick machine that selects a random number without replacement, i.e., that explicitly avoids giving duplicates. But of course, the decision to use such a machine is a cooperative act which is merely facilitated by a machine.
Second, why is there no profit until at least \(\$583\) has been bet? The reason is subtle, but informative. Consider for a random ticket, the probability distribution of tickets held by the crowd. These are
Table (ref) shows the probabilities and the syndicate's expected winnings as a function of the crowd holding $4$ or fewer winning tickets. The first column shows the number of winning tickets ($k$) held by the crowd, the second, the probability $P[K=k]$ that the crowd holds $k$ winning tickets, the third and fourth, the amount won and the $k$-contribution to the expected payoff ({$E[W_{1000}]$) for a syndicate that bets $s=1000$ different tickets, respectively, and the fifth and sixth, the amount won and the $k$-contribution to the expected payoff ({$E[W_{1}]$) for a syndicate that bets $s=1$ ticket, respectively.
A syndicate that bets only $\$1$ has expected payoffs of only $\$0.368$, $\$0.184$, etc. for a total of $\$0.632$ as shown in column (6). This is short of the syndicate's \$1 bet by $\$1 - \$0.632 = \$0.368$ which is the expected amount that goes into the carryover pool (and approximately, the probability of $k=0$) when $1001$ tickets are bet at random. But a syndicate that bets $s = 1,000$ reduces the probability of a carryover to zero, so that the term $k=0$ contributes $0.73576$ per dollar bet compared to $0.368$ per dollar bet when $s=1$. Not until the syndicate bets $291$ are there enough tickets unbet by the crowd, that the marginal return on the next ticket is greater than zero. After $291$, the probability of betting a ticket the crowd does not hold increases with each additional $\$1$ and the expected return therefore increases monotonically beyond that number due to succesive reduction of the probability that there are no winners.
Using the notation of Section (ref), we study lotteries for which $v = s + c$, $p = t^{-1} 1$, $a = x = 0$, and $r = e_s$, where $e_s$ is defined in (ref) and (ref). Our main interest is the returns from various choices of $s$, $c$, $q$ and $r$: \[ R(s,c,r,q) = R(s,c,p=t^{-1} 1,r,q,a=0,x=0) \] These lotteries have most of the important characteristics of more complicated ones, but are easier to analyze. In the initial part, we assume that each group in the crowd consists of a single bettor. This condition is relaxed in Proposition (ref).
A lemma is useful for the main theorem. Its straightforward defivation follows that of (ref)-(ref) and is omitted.
The main result is
Proof: From Lemma (ref), we determine that
The functions
in equation (ref) are positive, strictly decreasing and convex on $(0,1)$ for $c \ge 2$ (Appendix (ref)). Since a sum of $t$ (strictly) convex functions on $(0,1)$ is a (strictly) convex function on $(0,1)^t$ and remains (strictly) convex on any convex subset of $(0,1)^t$, it follows that the sum (ref) is strictly convex on the simplex \[ \left\{ \: q \in (0,1)^t \; \bigg| \; q \ge 0, \, \sum_{i=1}^{i=t} q_i = 1 \right\}. \] Further, a constrained optimization of the sum in (ref) yields the first order conditions \[ - \frac{1 - (1 - q_i)^{c + 1}}{q_i^2} + \frac{(c + 1)(1 - q_i)^{c}}{q_i} - \gamma = 0, \] for constant $\gamma$ and $i = 1, 2, \ldots, t$. These conditions are satisfied for $q_i = 1/t$, which in view of previous remarks shows this to be the unique minimum of (ref) in the simplex. We have thus shown (ref):
with strict inequality when $q \ne e_t$, which proves inequality (ref).
For inequality (ref), $E[ R(s,c,e_s,q) ]$ as a function of $s$ on $[1,t]$ is linear with positive slope. Therefore, it achieves its unique maximum at $s = t$.
Since the syndicate's edge is minimized when $q = e_t$ (by (ref)) and its gain maximized when $s = t$ (by inequality (ref)), inequality (ref) will be demonstrated if
is true. But rearranging (ref),
To verify inequality (ref), let $g(x)$ be the strictly concave function \[ g(x) = log \left( \frac{t + x}{t} \right) \] and $X$ the random variable
Then using Jensen's inequality,
Multiplying both sides of (ref) by $c+1$ then gives (ref).
$\blacksquare$
Proof: Immediate from Theorem (ref).
$\blacksquare$
We address some further questions about the syndicate's expected return in the next two propositions.
Proof: The exact expected gain from (ref) is \[ g(s) = E[ G(s,c,e_s,e_t)] = \frac{(c+s)s}{c+1} \left( 1 - \left( 1 - \frac{1}{t} \right)^{c + 1} \right) - s \] for $s \in [0,1]$ when the crowd bets optimally using $q = e_t$. Setting $y = (1 - \frac{1}{t})^{c+1}$, we calculate the first derivative
and find a single critical point
Substituting $s^*$ into $g$ gives the minimum value
which is $< 0$. Since \[ g''(s) = \frac{2}{1 + c} (1 - y) > 0, \] $s^*$ is the unique minimum of $g(s)$ and $g$ is convex. The question is: does $s^*$ lie between $0$ and $t$? Since (ref) is clearly positive, we check for $s^* < t$. But this is clearly the case, since $g(0) = 0$, $g(t) > 0$, $g(s^*) < 0$ and $g$ is strictly convex on $[0,\infty)$.
Since $g(s)$ is strictly increasing on $[s^*,t]$, $g(s^*) < 0$ and $g(t) > 0$, there is a point at which the gain breaks even. Since $g$ is quadratic in $s$ and symmetric around $s^*$, it follows that $s_0 = 2s^*$ is the break-even point:
$\blacksquare$
Proposition (ref) shows that a syndicate's bet of an amount $s \le t$ will have decreasing expected gain for $0 \le s \le s^* = (1 + c y)/(2(1 - y))$, increasing expected gain for $s^* \le s \le t$ and will be positive only if $\floor{s_0} + 1 \le s \le t$.\footnote{ $\floor{y}$ is the floor function, the greatest integer not greater than $y$.}
Betting groups at lotteries are either single individuals who bet a block of different tickets or collections of bettors who pool their funds and act as an individual by selecting different tickets. In either case, a group bets different tickets. In order to simplify the analysis, we assume that there are $g$ groups that bet the same number $l$ of different tickets, so that $c = g l$. Assuming that the crowd aligns their betting to the lottery probabilities, the $i^{th}$ ticket will be selected by a group with probability $l/t$ and therefore the number of tickets bet on the $i^{th}$, $K_i$, is distributed binomially: $K_i \sim Bin( g, l/t )$.
Proof: It is shown in Appendix (ref), formula (ref) that
when $c \gg 1$ and $X \sim Bin(c,q)$. Thus
When $l = 1$ (so that $g = c$) we get the usual calculation of expectation, and the ratio is
But under the assumption $l \ll min(c,t)$, (ref) will be very close to 1 since
$\blacksquare$
Of course, this approximation breaks down if some of the groups have sizes comparable to the syndicate; it would seem that the syndicate's gain is more tied to $\underset{1 \le j \le g}{max} \, c_j$ than an average of the $c_j$, but this idea is not investigated here.
The proof is in Appendix (ref).
There is an important message in part (ref) of Theorem (ref) --- that if $n > 1$ syndicates all bet each ticket once in a lottery with no take and no carryover pool, then each syndicate's expected return is (still) positive!
In this section, it is assumed that true ticket probabilities have a distribution $P[ D = i] = p_i$, where in general $p \ne e_t$, that the crowd bets using probability vector $q$, the syndicate bets using $r$, and $a, x > 0$. In this section, $s r$ will generally be a fractional vector, since that allowance produces tractable solutions. In practical usage, though, these fractional solutions must be converted into integral ones, and then examined to ensure that they retain near-optimal properties.
All results of this section apply without modification to equiprobable lotteries. The winners of a lottery will share the jackpot pool of $v = a + (1-x)(s + c)$, where $a \ge 0$ is a carryover pool, $x$ is the take on the betting pool, $s$ is the amount of the syndicate's bet and $c$ the amount of the crowd's. As in Section (ref), let each crowd member select one ticket independently of everyone else resulting in a random selection $K = (K_1, K_2, \ldots, K_t)'$, $\sum_{i=1}^{i=t} K_i = c$, where $K_i$ the total number bet on the $i^{th}$ ticket. Then the random variable $K$ has a multiomial $Multi(c,q)$ distribution \[ P[K_1 = k_1, K_2 = k_2, \ldots, K_t = k_t] = \frac{c!}{k_1! \, k_2! \, \ldots, k_t!} q_1^{k_1} \, q_2^{k_2}, \ldots, q_t^{k_t}. \] and the marginal distribution $K_i$ of $K$ is binomial with probability $q_i$: $K_i \sim Bin(c,q_i)$
The contest between syndicate and crowd can be considered as a game in which the probability distribution for tickets is $p$ is known to everyone, the syndicate bets according to $s = s \cdot r$, and the crowd independently according to $K \sim Multin(c,q)$, where $Multin$ is a multinomial distribution. The game commences with syndicate and crowd selecting tickets, with members of the crowd independently selecting 1 ticket apiece. After ticket selection, a random winning ticket is drawn according to $p$. The payoffs of this game for the syndicate and crowd are their respective expected returns from strategic choices $(r,q)$.
A Nash equilibrium in this game consists of strategies $(r,q)$ for syndicate and crowd such that given $r$, $q$ is a best returning strategy for the crowd and given $q$, $r$ is a best returning strategy for the syndicate. We show below that when $c,s \rightarrow \infty$ and $c/s \rightarrow u$, $u$ a constant, it follows that $r = p$ and $q = p$ asymptotically. Our definition of risk aversion and risk seeking differ from that standard in game theory and economics and is motivated by the game we have defined, in which the crowd as a whole is considered risk averse or risk seeking since in effect, we treat the crowd as a single stochastic bettor.
Proof: Part (ref). Suppose that fractional tickets can be bought, and that the syndicate purchases $s$ tickets in fractional amounts $s_i = s p_i$ and let $D$ be the winning lottery ticket. Since $D$ has distribution $P[D = i] = p_i$, the fractional number of tickets bet by the syndicate is $s_{\!_D} = s \, p_{\!_D}$ and the random variable for the crowd's bet is $K_{\!_D}$ with distribution $K_{\!_D} \sim Bin(c , q_{\!_D})$. With
the syndicate's expected gain is
where the last step follows from Jensen's inequality. It can be shown using constrained optimization that expression (ref) is minimized with respect to $q$ at $q = p$. Therefore
Part (ref). Consider first a lottery with no take in which the syndicate bets proportionally to $p$ and the crowd uses probabilities $q$. What is the best asympotic choice of $q$ to minimize syndicate's expected gain. We calculate $E[G(s,c,p,r,q)]$ as
The first order conditions require that an optimum $q_i^*$ satisfy
for some constant $\beta$ for all $i = 1, 2, \ldots, t$, where
If the lottery is equiprobable ($p_i = 1/t$), then $q_i^* = p_i = t^{-1}$ for all $i$. If the lottery is not equiprobable, then for at least two tickets $i$ and $j$, $p_i \ne p_j$ and from (ref), $q_i^* \ne q_j^*$ and the equiprobable argument does not work. In this case, the best choice for $q^*$ will in general not be $p$.
But if $c, s \rightarrow \infty$ and $c/s \rightarrow u$, $u$ a constant, then
and a Lagrange optimization of $q$ in (ref) yields the equations
for a constant $\beta$ and for each $i = 1, 2, \ldots, t$. These equations can be constant only if $q_i / p_i$ is constant for each $i$ which implies $q_i = p_i$.\footnote{Setting (ref) equal for $i \ne j$ yields the unique solution $q = p$.} The Hessian from (ref) has positive entries on the diagonal and zeroes off the diagonal, therefore is positive definite; thus $q = p$ uniquely minimizes the syndicate's gain. Conclusion: Choosing $q_i = p_i$ for each $i$ minimizes the syndicate's asymptotic expected gain.
Now suppose that the crowd bets using $p$ and the syndicate bets proportionally to $r$. Then
Then
and the first order conditions from equation (ref) can be written
and using the same argument as earlier, it follows that $r$ = $p$. Taken together, these results show that, asymptotically at least, the unique best reply of the crowd to $r = p$ is $q = p$ and the unique best reply of the syndicate to $q = p$ is $r = p$, which is precisely a unique Nash equilibrium.
Part (ref). The general syndicate expected gain is
The problem is: What distribution $p$ maximizes $ G(s,c,p,r,q)$ given $q$, $s$ and $c$. In this case no calculation is necessary to get the answer. This is because the first order equation for each $p_i$ must equal the same constant $\beta$, and selecting $p = (1/t) 1$ solves this problem. The function \[ \frac{z}{z + x} \] is concave in $z$, so $p = (1/t) 1$ is the unique maximum.
Part (ref). A crowd's risk seeking is underbetting of safe bets and overbetting of long shots (the favorite-longshot bias) and a crowd's risk aversion is the reverse. We may express this as follows: let the probabilities $p_1$, $p_2$, \ldots $p_t$ be ranked from highest to lowest \[ p_{(1)}, p_{(2)}, \ldots, p_{(t)}. \] A crowd is risk seeking if
where \[ \sum_{i=1}^{i=t} q_{(i)} = 1, \] $u$ is nondecreasing, $u_j \ge 0$, $u_{(1)} < 1$, and $u_{(t)} > 1$.
An analogous definition for risk averse requires that $u$ is nonincreasing, $u_j \ge 0$, $u_{(1)} > 1$ and $u_{(t)} < 1$ but such behavior is seldom exhibited at racetracks.
But we are done, since from part (iii) we know that such strategies are asymptotically worse than proportional strategies.
$\blacksquare$
The main result of this paper is that a single syndicate has a mechanical strategy that achieves excess returns against a crowd of uncoordinated bettors when (1) lotteries are equiprobable and have no take (Section (ref)) or (2) lotteries having jackpots and known probabilities satisfy a condition involving the carryover pool, the lottery take and the size of the betting pool. (Section (ref)).
There are two reasons for these excess returns: convexity of payoffs and a failure to cooperate. Regarding convexity, if payoffs were linear then arguments fail because applications of Jensen's inequality will yield equalities. Regarding non-cooperation, if everyone cooperates then there is one large syndicate that does not have a positive expected return.
Economics explains this sort of behavior by asserting that crowds at racetracks and in lotteries act “rationally” using diverse utility functions. And the difficulty of reconciliating those utilities along with a desire (incorporated into them) “to win the big one” without sharing with others, promotes non-cooperation. Of course, this behavior fits nicely with economic theory, but in the process creates an opportunity for substantial returns.
In any case, the rational actor model is difficult to defend as emphasized by Daniel Kahneman:
Kahneman's behavioral finance explanation leads to a different model of lottery behavior: emotional arousal overwhelms rational, calculated weighing of risks and rewards. The consequence is clear --- “irrational” crowd betting will persist, since changing emotional responses is very, very difficult. Moreover, persons motivated by irrational lottery-itis have a behavioral incentive not to join syndicates, because that eliminates the excitement. Therefore, for this inefficiency not to be exploitable, there need to be significant “limits to arbitrage.” An example explains one limit to arbitrage in government lotteries:
We have demonstrated that there generally exists a purely mechanical strategy that produces excess returns in pure jackpot lotteries. When the lottery has no take and no carryover pool, the edge of a syndicate is reduced, but not eliminated if other syndicates also participate (Section (ref)). We showed that for lotteries having take $x$, carryover pool $a$, and syndicate and crowd bets of $s$ and $c$, respectively, a single syndicate has an edge if $a/(t+c) - x \ge 0$. But a competing syndicate can convert an apparently favorable bet into an unfavorable one if $a/(2t+c) - x < 0$.