Then we analyze a truncated game of the firm against a single investor and show that this dynamic game has two. However, this kind of game is a misspecification of the actual interaction between the population of individuals and the firm under incomplete information. In Section 3, we build a parametric model of the Ponzi game and define the optimum strategy of the firm.

## PONZI GAME: PARAMETRIC EXAMPLE AND STRATEGY OF THE FIRM

Let nt–1 denote the number of company investors (I-strategists) at the beginning of period t. 12 This is, of course, a variant of the standard stopping condition in the theory of the firm.

## INDIVIDUAL STRATEGIES AND EQUILIBRIUM ANALYSIS A. Two kinds of individuals

Accordingly, the firm's strategy also determines the firm's optimal stationary policy in the stochastic game Γ. The optimal stop time thus depends on the company's expectations of the nt dynamics to which we are now moving.

## Strategies of sophisticated individuals in a dynamic game Under the settings specified above, the equilibrium analysis of the Ponzi

In contrast to naive investors, they have heterogeneous beliefs about firm demise, which also vary over time, as indicated by the appropriate probability measure. In other words, these people realize that investing in such a company is very dangerous, but they are even more convinced that they are smarter than the company and will withdraw before the plan collapses. An individual who believes that the company will cooperate must play in phase game t with a probability greater than 1/(1 + d) I; and if the firm is of type θ+t >θ∗t, it will continue to cooperate at t, so that each (θt+C, I) profile forms an equilibrium.

The next formal result, proven in the appendix, arises from the fact that if the firm's intention to deviate from the staged game at t is not widely known, individuals can have a set of consistent beliefs about the firm's strategy the company, some of which it may be true. In the truncated phase game in strategic form in Table 2 (as part of the Ponzi game), the equilibrium components (θt+C, I) and (θt−D, W) will contain perfect, successive, correct, stable and essential equilibria when the company deems it optimal to cooperate and fail, respectively, and the individual attaches sufficiently high probability to these events. As long as the firm's intention to defect coincides with the individual's intention to quit, the profile would be '(C, I) at each stage prior to T and (D, W) at T' in this truncated game' look like this'. a balance that will also be sequential, perfect, subgame perfect and Markov perfect.

A cautious strategy is probably optimal in an extension of the truncated game; but when an individual plays against a company, the company is not only playing against that individual, but against the population. Policies ψ5 that maximize this payoff represent the truly best response in the Ponzi game.

## Equilibrium analysis of the Ponzi game The considerations provided above led to the following proposition,

### EVOLUTIONARY DYNAMICS OF INDIVIDUAL BEHAVIOUR The above analysis was essentially Bayesian in its spirit; thus, it restricts

Now is the right time to continue with an explicit description of both processes, which leads to a key component that we have omitted so far - the dynamics of net investor inflows.

## Naive individuals We call naive individuals boundedly rational in the sense introduced in

All these models stipulate an increase with time of the share of better equipped strategies within the population. Finally, expected payoff of a randomly selected member of the population characterized by mixed strategy q, versus an opponent playing mixed strategy p, is denoted by u(q,p). Our work is along the same line of research, which proposes an economic application of some generalizations of the dynamics (5).

In our case, the vector with such proportions q = [qI, 1 - qI] = [qI, qW] can be formally associated with the mixed strategy of the population. Below we construct a variant of the discrete-time deterministic replicator dynamics (adapted from Weibull, 1995) which is explicitly derived from individual adaptation strategies. Specifically, we assume that revisions of a current strategy for each member j of the subpopulation of naive subjects follow a Poisson process with arrival rate rj and each individual j switches to strategy i with probability π ∑π.

It assumes that the revision rates are constant across a population (set ri = 1, ∀i), but the probabilities of switching to the strategy of randomly selected individuals depend on the relative fitness of the two strategies. Under this assumption, a discrete-time version of the replicator dynamics can be obtained from (11) as.

## Sophisticated individuals The model considered in the previous section dealt exclusively with naive

It is natural to assume that the longer the pyramid exists, the lower the sophisticated I-strategist's confidence that the firm will not defect in the next period. Alternatively, the same previous vector can be multiplied by the t-step transition matrices calculated via the Chapman-Kolmogorov equations, i.e., the matrix of transition probabilities after two steps is as shown in table 5. If ϖ0 - the value of low of μ - is arbitrarily close to 0 (corresponding to those investors who are rational in the Bayesian sense, but do not think that the firm is likely to defect at all), then ϖt - the minimum of posterior beliefs at t - will remain close to 0, and the behavior of the holder of such beliefs will resemble that of naive individuals.

An assumption of a uniform distribution of beliefs in (ϖt, wt), together with a continuous approximation of beliefs and risk neutrality, leads to the following sequence of posterior probabilities for the entire class of sophisticated individuals: 13) This expression determines the probability that an arbitrarily chosen sophisticated individual prefers I to W. Another component of our construction applies to those sophisticated individuals who were playing W because of their ignorance of the pyramid's existence, but became aware of it together. with its growth. That way, at any time period in the early phase of the pyramid growth there will be a uniform.

At t→∞, it will happen that d/(1 + d) <ϖt, which means that the least skeptical individual will not invest and the share of newcomers will become zero.22. 22 We safely rule out the case w0> d/(1 + d): if sophisticated individuals are "too skeptical" to invest, their behavior is irrelevant to the growth of the pyramid.

## The dynamics of pyramid's growth Now we may explicitly combine the two dynamics — (12) for naive and

### NUMERICAL SIMULATIONS AND ALTERNATIVE SPECIFICATIONS

Empirical tests of the above dynamic models are rather difficult, because no data on the pyramid's growth is readily available. The only publicly available information is the number of "diluted debt holders," n*, the maximum amount of capital k*, and also the pyramid's lifetime. For example, according to Russian media, one of the most famous Russian pyramids (Khoper-Invest, Rostov-on-Don) attracted 1500 billion rubles from about 2.5 million private investors; another (Russian House Selenga, Volgograd) — 2800 billion rubles.

## Basic dynamics We used dynamics (20) to simulate pyramid growth with different values

Not surprisingly, a higher value of c leads to a decrease in n* and k*, as well as an increase in the pyramid's lifetime — in our estimates we used c = 0.1. The dynamics of the pyramid's growth were also not sensitive to the initial value of capital, implying that the modern Ponzis firm could start with virtually no fixed costs, and that public trust was the main source of its success. The main determinants of the dependence of these dynamics on time are the unobservable parameters a and b, which had to be adjusted.

At intermediate levels of b at about 0.010 to 0.025, the dynamics of the proportion of I-strategists reaches a steady state that depends on b; initially qI exceeds this equilibrium state, but gradually qI returns to it from above. Also of interest is the question of whether the composition of the population is important (i.e. the proportions of naive and sophisticated individuals in N). When most subjects are sophisticated, population dynamics closely mimic those of their subpopulation (Figure 3).

However, our model reveals some of these psychological characteristics: thus, as shown in Fig. 4 and 5, the entire story of the pyramid is not necessarily over at the maximized value of capital. This characteristic of the replicator dynamic captures the fact that it takes time for the population of naive individuals to accept the end of their dream related to the pyramid.

## Advertising campaign The first obvious extension of the baseline model (20) consists of the use

Even after the firm is destroyed (kt < 0), the proportion of its naive investors does not immediately fall to zero. Lower values of d moderate the growth of the pyramid and increase the lifetime of the pyramid with higher b, but drastically reduce its size. Since very high levels of d will lead to explosive dynamics and may not seem reliable, moderate values of d (say, 10 to 20% above the market rate) seem optimal and were indeed the most common in reality.

In both cases, the smoothing of the dynamics occurs at a rather high cost; a less expensive way to achieve the same goal will be discussed in the next subsection. In such a case, the higher this increase is in the last period, the higher the amount allocated for advertising will be. Dividing by kt in (21) is again a matter of normalization, which also corresponds to the fact that the higher kt, the lower the firm's incentive to advertise further.

Symptomatically, both lines are the same at the beginning and only slightly decrease at the top of the pyramid, corresponding to the fact that at higher b advertising incentives decline, but only at higher values of kt. First, the peak of investors and capital is reached faster, by period 5, which is synchronized for different values of b.

Effects of nonlinear utilities Another possible generalization of dynamics (20) arises if individuals

## A simpler derivation of replicator dynamics It has been noticed before that a strategically minded sophisticated

### CONCLUSION The issues discussed in this paper are pertinent to the puzzling and im-

Given the finiteness of the set of all possible histories, {kt} is a sequence of well-defined random variables — values of capital in each period. Proceeding recursively backwards, we see that the same strategy will ensure that Vt is an optimal value of capital for period t and all subsequent periods t + 1, .., T due to the enforcement of the optimality principle. We construct a complete type space for the initial stage game γ0 of the game Γ, followed by the gradual ones.

Assume that the capital of the company takes values on a certain subset of the positive half of the real line, denoted by K0, and that the initial strategy of the population of individuals is characterized by parameter vector W0. 27 The compactness assumption is hardly too strong in this context, because the set of possible strategies of the firm is finite (|Q| = 2), the population is finite, and possible values of the firm's capital are bounded from above by all points of view of individuals. This completes the construction of the universal belief and the space for the initial phase game γ0.

We can do this by noting that the universal belief space Ωt contains all possible paths of the games γt, t, where the observed part of the profile qt "cuts off" those paths (subsets of Ωt) that are known not to be played at each time period. We can say that each move by the firm divides the set of possible paths of the Ponzi scheme into two parts corresponding to its cooperation or defection. When µt+1 is the indicator function of the set Ht+1, this is a version of the conditional probability;30.

These sequences can go on indefinitely, although in practice they are interrupted by the company's first defection.