# airdrop_games__0a172a6a.pdf

Airdrop Games

Sotiris Georganas1,3 , Aggelos Kiayias2,1 and Paolo Penna1

1IOG 2University of Edinburgh 3City, University of London sotiris.georganas@iohk.io, kiayias@inf.ed.ac.uk, paolo.penna@iohk.io

Launching a new blockchain system or application is frequently facilitated by a so called airdrop, where the system designer chooses a pre-existing set of potentially interested parties and allocates newly minted tokens to them with the expectation that they will participate in the system such engagement, especially if it is of significant level, facilitates the system and raises its value and also the value of its newly minted token, hence benefiting the airdrop recipients. A number of challenging questions befuddle designers in this setting, such as how to choose the set of interested parties and how to allocate tokens to them. To address these considerations we put forward a game-theoretic model for such airdrop games. Our model can be used to guide the designer s choices based on the way the system s value depends on participation (modeled by a technology function in our framework) and the costs that participants incur. We identify both bad and good equilibria and identify the settings and the choices that can be made where the designer can influence the players towards good equilibria in an expedient manner.

1 Introduction Launching a new blockchain system is challenging as it requires the upfront contributions of different parties, without any guarantee that the system will be successful. The characteristics of such launches are as follows: There is a set of possibly interested parties. Participating incurs some cost, hence the system designer performs an airdrop of tokens to entice the participants: a certain amount of the available tokens are distributed in advance to potential contributors, regardless of their (future) individual contribution to the system [Allen et al., 2023].1 Identifying the potential contributors typically piggybacks on an existing blockchain system e.g., as in restaking in Ethereum where new tokens are allocated

1The nature of participation or contribution should be interpreted broadly and includes holding tokens, participating in governance, or actively running bespoke software that performs system functions.

based on existing staked ether holdings [Eigen Labs, 2014], but more direct approaches have also been attempted, e.g., in worldcoin [Worldcoin, 2025], prospective users scan their retina in order to receive tokens. The eventual success of the system depends on the actual contributions and level of participation, which, in turn, reflects on the monetary value of the tokens received via the airdrop. The higher the overall participation of the players, the higher is the value of the new token, and thus also the value of the airdrop allocation received initially. The dependency between system value and participation can be modeled by an underlying technology function that we make explicit below. Potential contributors thus face a dilemma: If they contribute, they incur a cost but (potentially) increase the value of their token allocation. Naturally, contributors should act strategically and contribute in a way that maximizes utility. Several equilibria exist: in good equilibria, enough participation is achieved and the launch of the system succeeds while in the bad equilibria a complete breakdown of the system is possible. From the designer s perspective, some fundamental questions need to be addressed in order to understand how a project can be successfully launched: What is the level of contribution of the parties that we can reasonably expect, given a specific allocation? How can this be influenced by different tokenomics policies that award larger or smaller amounts of tokens as part of the airdrop allocation? What kind of technology functions are more favorable in terms of facilitating a successful launch? In this work, we formally address these questions via a novel game-theoretic model and its analysis. To illustrate the nature of the problems, consider the following technology function: Example 1 (Threshold Technologies.). Consider a system technology that requires contributions from at least 50% of the contributors (the total number being N = 10). If this threshold is met, the system operates correctly, and the token s value is high, $10. Conversely, if the threshold is not reached, the system fails, and the token s value drops to low, at 0. With an airdrop granting each participant 1 token and contribution cost α = 1, two equilibria emerge: (i) no one contributes, since an individual contribution alone does not

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increase the token s value but incurs a cost of $1, and (ii) exactly 50% contribute, that is 5 players. If any one of them contemplated not contributing, that would cause the token s value to drop from $10 to 0 (a net profit change from 10 - 1= 9 to 0). Also none of the 5 players not contributing has any incentive to contribute, since they are already enjoying the high value without any cost. Clearly, the latter equilibrium is preferable, and its existence for higher costs is guaranteed only if the designer sets the airdrop properly (e.g., for participation costs of α = 20 USD, an individual airdrop of at least 2 tokens would be required).

The threshold-based technology described above is natural, but we can consider other types of technologies determining the system s value. For instance, [Alabi, 2017] suggests that several systems follow Metcalfe s Law: the value of a network is proportional to the square of the number ℓof contributors (number of entities holding the native token in a wallet). In this case, the token s value follows t(ℓ) = q ℓ2, with q a rescaling constant. The equilibria emerging with this technology are different from the example above, leading to different trade-offs when deciding on the number of tokens to airdrop.

1.1 Our Contribution

Our contributions can be summarized as follows.

Game-theoretic model (Section 2). We propose a gametheoretic model for airdrops. The model incorporates the key feature that token allocations are issued in some new token whose value: (i) is not determined at launch but, (ii) is affected by the actual participation or contribution of the potentially interested parties allocated these tokens according to some technology function. The analysis of equilibria in our model (see below) is informative to the designer and answers basic questions like Who receives tokens/how many? as posed in [Fr owis and B ohme, 2019]. Intuitively, the designer sets some eligibility criterion based on past information, which determines the number and the individual costs ( who receives tokens and why ), and the corresponding airdrop allocation ( how many tokens ).

Analysis of equilibria (Section 3). We characterize the set of pure Nash equilibria for the general setting, as a function of amount of rewarded tokens, the number of potential contributors, their individual costs, and the technology which converts individual contributions into system value. We show that the model s general version corresponds to a potential game (Theorem 1). Thus, pure Nash equilibria always exist and are reached via simple best response dynamics. We further characterize the set of pure Nash equilibria in Section 3.1. It is worth noting that (i) we consider heterogeneous costs, that is, players have different costs in general, (ii) pure Nash equilibria do not require players to know about others costs but only about others strategies (contributions), and (iii) these equilibria are quite natural as they arise from simple best response (as opposed to mixed Nash equilibria for which no simple dynamics exist, and whose empirical support is comparably limited at the level of individual player behavior in the lab or field). These considerations are also fundamental for the designer, as a target.

Refined unique (logit) equilibria (Section 3.2). For some technologies, bad equilibria where no player contributes coexist with good equilibria where a sufficiently high level of players contribution is reached, thus making the system valuable. We consider a well-known class of noisy best response dynamics, termed logit dynamics [Blume, 1993; Blume, 2003] (see Section 1.2 for further discussion) for our model. These seem natural in our context and do not require excessive sophistication from the players. We first show that, in the so-called vanishing noise regime, these dynamics select only stochastically stable pure Nash equilibria, characterized by Theorem 3, thus discarding bad equilibria in several cases (see below). We also consider the so-called finite noise regime and the corresponding unique stationary equilibrium [Auletta et al., 2011]. Under some mild restrictions on the model (Section 4), we provide tight bounds on the time for the dynamics to reach its equilibrium or a particular level of contribution (Section 4.1). Time is also a fundamental aspect for the designer, as the system needs to reach a sufficiently good state within the given launch period (after which the system is supposed to start its normal autonomous operations). Applications to relevant technology functions (Section 5). Our model accommodates generic technology functions to express the token value depending on the actual contributions of the tokens receivers. We apply our results to the important class of threshold technologies, a natural, simple description of systems which need an initial minimal base to succeed [Chaidos et al., 2023; Arieli et al., 2018; Yan and Chen, 2021; Jim enez-Jim enez et al., 2021; Chang, 2020; Shao et al., 2023; Wang et al., 2023]. Threshold technologies are a typical example where bad equilibria with zero participation always exist, along with good equilibria, so there is a need for theory to explain how successful systems can reach good equilibria. Threshold technologies represent a hard case in our setting, in the sense that bad equilibria persist for any airdrop reward amount. Logit dynamics provide a formal argument that good equilibria are more likely to be chosen (Theorem 6 and Corollary 2) also in this hard case. The analysis further indicates when it is optimal for the designer to perform an airdrop at all (and the optimal rewards) in both the vanishingand nonvanishing regimes (Sections 5.1 and 5.2). The analysis of threshold technologies highlights the need for low costs to converge to a desired state within reasonable time (Section 5.3). This highlights the benefits of recent technological developments like Ethereum s restaking [Eigen Labs, 2014] or Cardano s partnerchain framework [Ward, 2024], which reuse contributors from a mainchain who are likely to incur lower costs (see [Georganas et al., 2025]). We stress that our results can be applied to other technology functions, including Meltcafe s Law and other examples in the literature (details in [Georganas et al., 2025]).

1.2 Related Work Airdrops. Airdrops are costly as an action, as recording them on an existing chain incurs transaction fees, necessitating the designer to implement simple allocation strategies [Fr owis and B ohme, 2019; Lommers et al., 2023]. Empirical studies suggest simple airdrops will remain common

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for projects without established on-chain activity [Allen, 2024], while others [Messias et al., 2023] recommend scaling rewards with costs , aligning with our theoretical findings. Additional work on practical features and goals of airdrops includes [Makridis et al., 2023; Yaish and Livshits, 2024; Fan et al., 2023; Lommers et al., 2023].

Related Games and Models. Our model can be seen as a variant of blockchain participation games [Chaidos et al., 2023] and the combinatorial agency model [Babaioff et al., 2012] in contract theory. In the former, players receive a monetary reward, contrary to our setting where the rewards are tied to the system value (the variant of universal payments with no retraction is the closest to ours, whereas in other variants rewards are even more loosely tied to the system s success); the system value is a threshold technology of (eligible) players actively contributing. In the combinatorial agency model, players receive again a monetary reward conditioned on the success of the project, expressed by some success probability function on the contributing players. Consequently, the equilibria in these two models are different from ours and these results do not apply to our setting. A closely related class of problems is crowdfunding games [Arieli et al., 2018; Yan and Chen, 2021; Jim enez-Jim enez et al., 2021; Chang, 2020; Shao et al., 2023; Wang et al., 2023]. Similar studies concern public funding projects [Soundy et al., 2021; Bil o et al., 2023] and public goods projects on networks [Bramoull e and Kranton, 2007; Galeotti et al., 2010; Dall Asta et al., 2011; Yu et al., 2020]. These models are mathematically equivalent to our games in the case where the designer cannot change the rewards. Specific problems correspond to technology functions being S-shaped [Buragohain et al., 2003] or a threshold function [Galeotti et al., 2010]. Stochastic stability is studied in [Boncinelli and Pin, 2012]. All these problems (and ours) belong to rich class of public goods games. These are generally computationally hard, even when the underlying (technology) function is specified by a network [Papadimitriou and Peng, 2021; Gilboa and Nisan, 2022; Klimm and Stahlberg, 2023; Do Dinh and Hollender, 2024; Galeotti and Goyal, 2010].

Logit Dynamics. Logit dynamics have been largely studied in the context of games and equilibrium selection problem, that is, as a refinement of pure Nash equilibrium (see e.g. [Blume, 1993; Blume, 2003; Montanari and Saberi, 2009; Asadpour and Saberi, 2009; Al os-Ferrer and Netzer, 2010; Al os-Ferrer and Netzer, 2015; Auletta et al., 2011; Auletta et al., 2012; Auletta et al., 2013b; Auletta et al., 2013a; Okada and Tercieux, 2012; Coucheney et al., 2014; Ferraioli et al., 2016; Mamageishvili and Penna, 2016; Ferraioli and Ventre, 2017; Al os-Ferrer and Netzer, 2017; Penna, 2018; Kleer, 2021]). This is usually done in two ways. The first is to consider the so-called vanishing noise regimes and a resulting set of stable equilibria (see e.g. [Blume, 1993; Al os Ferrer and Netzer, 2010; Al os-Ferrer and Netzer, 2015]). The second is to consider non-vanishing noise regimes and the corresponding unique stationary distribution of the process as the equilibrium concept [Auletta et al., 2011]. The logit response model also finds applications in economics [Costain and Nakov, 2019], in pricing algorithms [M uller et al., 2021;

van de Geer and den Boer, 2022], and coalitional bargaining [Sawa, 2019].

2 Modelling Airdrop Games The model captures the following key aspects of airdrops: (i) The designer chooses the amount of tokens to be airdropped to potential contributors. (ii) Potential contributors decide whether to perform a (costly) task for the system. The resulting system value depends on the total contribution and the underlying technology of the project. (iii) Contributors maximize utility, resulting in an equilbrium and system value. The designer faces a tradeoff between the airdrop amount and the resulting system value (too small airdrops do not incentivize enough contributors and thus the project fails, while giving away all tokens is not optimal either because it minimizes profit).

Parameters and Underlying Game Contributors. There is a set of n potential contributors (players): Each contributor chooses her actual contribution (strategy) ai Ai R+, incurring in a cost of ci ai where ci denotes the per unit cost of i. System (technology) value. The overall value of the system depends on each individual effort or contribution, i.e., on the strategy profile a = (a1, . . . , an) and it is equal to V (a) for some monotone non-decreasing function (higher contributions yield the same or higher value). Token value (and total supply). Given the token total supply (the overall number of tokens of in system) Ttot, the value or price of the token is

t(a) := V (a)

Airdrop (Token) Allocation. The designer allocates some constant fraction ρ [0, 1] of the overall token supply as an airdrop, i.e., to be distributed equally among the players. Thus, each player receives γ tokens, where

γ := ρ Ttot

n ρ [0, 1] . (2)

(For the sake of simplicity, we allow γ to be a fractional number, and ignore rounding effects.) It is worth noting that the monetary reward (number of tokens times the token price) is independent on the token total supply Ttot, and it equals to a fraction ρ/n of the system value,

γ t(a) (1)+(2) = ρ Ttot

n V (a) . (3)

Utilities. Players utilities equals the monetary reward received (number of tokens times the token value) minus the incurred cost

ui(a) := γ t(a) ci ai (3)= ρ

n V (a) ci ai . (4)

Equilibria. A strategy profile a = (a1, . . . , an) is a pure Nash equilibrium if no player i can increase her utility by changing her strategy ai, that is,

ui(a) ui(a ) (4) ρ n (V (a) V (a )) ci (ai a i)

(5) for all i and all a = (a1, . . . , ai 1, a i, ai+1, . . . , an).

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Logit dynamics. Logit dynamics [Blume, 1993; Blume, 2003] are a kind of noisy best response dynamics where players have some inverse noise or learning rate β 0 and each of them responds according to a so-called logit response

pβ i (ai|a i) = exp(βui(ai, a i))

where Zβ i is a normalizing factor so that the above formula is a probability distribution, and (x, a i) := (a1, . . . , a i, x, ai+1, . . . , an). For β = 0, players choose a strategy at random with uniform distribution, while for β , they tend to the best response rule.2 At each step of the dynamics, a randomly chosen player revises her current strategy according to the logit response above (6). Logit dynamics converge to a unique equilibrium πβ given by the stationary distribution of the underlying Markov chain: πβ(a) is the probability of players selecting profile a after sufficiently many steps of revisions ( learning ) have been performed. Note that this depends on the parameter β. For the class of exact potential games, in the vanishing noise regime (β ), equilibrium πβ concentrates on the subset of pure Nash equilibria whose potential is optimal [Blume, 1993; Blume, 2003]. The mixing time of the underlying Markov chain is the time required by the dynamics to reach the equilibrium πβ starting from any state [Levin et al., 2006; Auletta et al., 2011].

Objectives and Metrics There are different (possibly conflicting) metrics to evaluate system performance, given contributions, costs, the designer s profit etc. System Value. This is the value specified by the technology function V (a) as a function of all contributions. Social Cost. This is the sum of all players costs,

SC(a, c) := X

i ci ai . (7)

Users Welfare. This is the sum of all players utilities (4),

UW(a, ρ, c) := X

i ui(a) (4)+(7) = ρ V (a) SC(a, c) . (8)

Profit. This is the value of the remaining tokens remaining with the designer, after subtracting the airdropped tokens and the cost d V for developing the technology V ( ),

profit(a, ρ) = (1 ρ) V (a) d V . (9)

The designer strategically chooses the airdrop allocation ρ aiming to maximize its profit (utility) defined as in (9). 3 We consider which system values can be achieved given the underlying technology, players costs, and their utility maximizing strategies. Note that the designer s profit can be negative (also when the former does not provide any reward, ρ = 0).

2In case multiple best response exist, the corresponding player chooses any of them with uniform distribution. 3Note that we still assume the designer to move first by announcing the airdrop ρ, and then the (other) players will reach some equilibrium accordingly.

This corresponds to inherently bad projects that are destined to fail and should not be implemented. More generally, we consider a technology implementable or profitable if there is some equilibrium (also in randomized sense Section 3.2) which yields a positive profit to the designer.

Special Cases We shall sometime consider the following relevant restrictions on the technology functions, the possible contribution levels, the possible costs, and combinations thereof. Anonymous Technologies. It is natural to assume that the value of the system simply depends on the overall level of contribution ℓ= P i ai, that is, V (a) = V (a ) whenever P i ai = P i a i. We refer to this case as the anonymous technology function. With slight abuse of notation, we write V (ℓ) instead of V (a). Binary Contributions. In some settings where potential contributors have only two options (strategies), either to contribute (ai = 1) or to not contribute (ai = 0), we refer to this restriction as binary contributions. Uniform Costs. It is natural to consider equal cost for all (e.g. if they are fully determined by the type of hardware/resources required). This means ci = α, with α > 0.

3 Airdrop Games: Main Characteristics In this section, we consider games with utilities in (4), airdrop allocations (2) in full generality. We show that these games are always potential games (Theorem 1), implying that (i) best/better response dynamics always converge to pure Nash equilibria and (ii) logit dynamics equilibria can be characterized in terms of potential, implying that bad equilibria are not selected under vanishing noise (Section 3.2). Theorem 1. For airdrop allocation (2), the game in (4) with arbitrary efforts and any technology function is an exact potential game with potential function

ϕ(a) := γ t(a) SC(a) = ρ

n V (a) SC(a) . (10)

3.1 Characterization of pure Nash equilibria The next theorem characterizes the set of pure Nash equilibria. Intuitively, airdrop allocations should be neither too high (otherwise players can benefit by increasing their contribution) nor too low (otherwise they can benefit by reducing their contribution). Note an important asymmetry between the two cases: increasing the contribution is never beneficial when the system s value does not change, while decreasing the contribution is always advantageous under these conditions. Theorem 2. For any technology function and arbitrary efforts, and for airdrop allocations (2), the set of pure Nash equilibria is given by the strategy profiles a such that the following two conditions hold for all i: 1. For all a+ = (a+ i , a i) with a+ i > ai:

ρ n ci a+ i ai V (a+) V (a) whenever V (a) < V (a+) .

2. For all a = (a i , a i) with a i < ai:

ρ n ci ai a i V (a) V (a ) and V (a) > V (a ) .

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The next corollary provides a more convenient characterization for certain restrictions of interest. Corollary 1. For any technology function and binary efforts, and for airdrop allocations (2), the set of pure Nash equilibria is given by the strategy profiles a such that the following two conditions hold: 1. For all i such that ai = 0: ρ n ci V (1,a i) V (a) or V (1, a i) = V (a) . 2. For all j such that aj = 1: ρ n cj V (a) V (0,a j) and V (0, a j) < V (a) . Moreover, for any anonymous technology function and binary efforts, the set of strategy profiles a which are an equilibrium corresponds to those satisfying these two conditions:

1. For ℓ< n: ρ

n c(0) min V (ℓ+1) V (ℓ) or V (ℓ+ 1) = V (ℓ) .

2. For ℓ> 0: ρ

n c(1) max V (ℓ) V (ℓ 1) and V (ℓ 1) < V (ℓ) .

where c(0) min = min{ci : ai = 0} is the smallest cost among players not contributing, and c(1) max = max{ci : ai = 0} is the largest cost among players contributing. Example 2 (linear technology with heterogeneous costs). For linear technologies, V (ℓ) = λV ℓwith λV > 0, the set of pure Nash equilibria is characterized by the ℓ such that (w.l.o.g. assume c1 c2 cn) these inequalities hold: cℓ λV ρ n cℓ +1

λV . The optimal strategy (profit maximizing) for the designer is to choose the minimum ρ satisfying the above condition, thus making the first inequality tight: ρ = argmaxℓ [n](1 ρ ℓ) λV ℓ, ρ ℓ= n cℓ

λV , which is equivalent to set ρ = argmaxℓ [n](λV n cℓ) ℓ.

3.2 Logit response equilibria We next consider logit dynamics and the corresponding equilibria. Specifically, for the vanishing noise regime (β ), the dynamics selects a subset of so called stochastically stable pure Nash equilibria (see e.g. [Asadpour and Saberi, 2009]). Theorem 3. For any technology function and arbitrary efforts, for airdrop allocations (2), and for vanishing noise (β ), the dynamics converges with probability which tends to 1 to states of maximal potential. 4 In particular, the corresponding stationary distribution πρ, depending on the airdrop allocation ρ, satisfies

lim β (πρ(a)) =

( 1 |POTMAXρ| for a POTMAXρ 0 for a POTMAXρ (11)

where POTMAXρ := argmaxa{ϕ(a)} is a subset of equilibria which depends on the airdrop allocation ρ as follows:

POTMAXρ = argmaxa n ρ

n V (a) SC(a) o . (12)

The next example shows that, the bad equilibria in which no player contributes are selected with vanishing probability, provided the airdrop allocation ρ is sufficiently high.

4Note that in this work we do not change sign in the definition of exact potential game and related dynamics.

Example 3 (rule out bad equilibria). We consider a twoplayer game with a simple anonymous technology function, binary contributions, and uniform costs (ci = α). The technology function is an AND technology where a high (non-zero) value is achieved only if both players contribute, V (0) = V (1) = 0 and V (2) > 0, which implies that (0, 0) and (1, 1) are the only two Nash equilibria. Theorem 3 says that logit-response dynamics with vanishing noise (β ) the good equilibrium (1, 1) is reached with probability tending to 1 if and only if the airdrop allocation ρ satisfies ρ

2 (V (2) V (0)) > 2α. Also note that, since ρ 1, this holds only for V (2) V (0) > 4α. To repeat the intuition here, when contributors are prone to some experimentation, instead of just picking best responses, the system is likely to end up in the high value equilibrium instead of the low one.

Profit and optimal airdrops revisited (logit dynamics) We observe the same tradeoff regarding the optimal choice of airdrop ρ. A too small ρ may result in a bad equilibrium in which none cooperates, thus a system with small value t(0). A very high ρ, on the other hand, will leave the system designer with only a tiny fraction of the system ownership (value). In order to deal with the logit (randomized) equilibrium πρ we simply consider the expected value of the system, and extend the definition of profit (9) in the natural way:

profit(ρ) = (1 ρ)V (πρ) d V , where V (πρ) := E a πρ[V (a)] . (13)

4 Binary Efforts and Uniform Costs We consider a binary effort scenario where players contribute or not (ai {0, 1}) and costs are uniform (ci = α for all i). Under these restrictions, our model is characterized by three parameters: (i) α is the cost per player when contributing; (ii) β is the rationality level of the players; (iii) ρ is the airdrop allocation the corresponding number of tokens γ is given by (2). While the rationality level β is exogenous to the system, the designer can change ρ, and costs α are part of the airdrop design (see [Georganas et al., 2025] for practical examples where the designer can reduce α).

4.1 Logit equilibria and convergence time In this section, we make use of birth and death processes to analyze the case of binary effort and airdrop rewards, when also the technology function is symmetric. In this case, the dynamics boils down to a birth and death process, where we keep track of the number ℓof actively participating players in a given profile a, i.e., ℓ= P i ai and ai {0, 1}. Hence, the birth and death process has n + 1 states, ℓ [0, n]. The stationary distribution is thus

ˆπ(ℓ) := n ℓ

where π(ℓ) is the stationary distribution of a generic state a with P i ai = ℓ, and the binomial coefficient counts the number of such states in the original Markov chain which are grouped together in the birth and death process. The birth

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and death process has transition probabilities p(ℓ) and q(ℓ) of moving up by one or down by one , respectively, given by

n pβ i (1|a i) , q(ℓ) = ℓ

n pβ i (0|a i) (15)

where pβ i ( ) is the logit response (6). Since our process in (15) is an irreducible birth and death chain, the following sharp bound on the mixing time holds. Theorem 4 (Theorem 1.1 in [Chen and Saloff-Coste, 2013]). Let ℓ0 be a state satisfying ˆπ([0, ℓ0]) 1/2 and ˆπ([ℓ0, n]) 1/2, where ˆπ(I) := P ℓ I ˆπ(ℓ), and set

Tcutoff = max

ˆπ([0, ℓ]) ˆπ(ℓ)p(ℓ),

ˆπ([ℓ, n]) ˆπ(ℓ)q(ℓ)

Then the mixing time of the logit dynamics satisfies Tmix = Θ(Tcutoff) and, in particular, the following bounds hold:

(1/24) Tcutoff Tmix 288 Tcutoff . (17)

Note that such an ℓ0 always exists (sum up all ˆπ(ℓ) from 0 until the smallest ℓ0 where the sum of these probabilities is at least 1/2). We next introduce useful definitions to analyze the hitting time of a target ℓ, based on the technology s local steepness . Definition 1. We define the drift at location ℓas the ratio d(ℓ) := ˆπ(ℓ+ 1)/ˆπ(ℓ). We also say that technology function V is s-steep at some interval I = [ℓ1, ℓ2] if V (ℓ+1) V (ℓ) s for all ℓ I \ {ℓ2}. Intuitively, the drift describes the tendency of the process to move down (drift < 1) or move up (drift > 1). The following theorem states that the hitting time for a target value ℓ grows exponentially with the length of any interval, preceding the target value, where the tendency to move down persists. The theorem further connects the drift to the flatness of the technology function (see the threshold function in Section 5). Its proof is based on bounds in [Palacios and Tetali, 1996]. Theorem 5. The hitting time Thitting(ℓ) of the logit dynamics to reach a state with contribution level ℓstarting tom the state with contribution level ℓ= 0 can be bounded as follows: 1. If the drift in some interval I = [ℓ1, ℓ2] is at most d I, then Thitting(ℓ) (1/d I)|I| for all ℓ> ℓ2. 2. If V is s-steep in some interval I = [ℓ1, ℓ2], then for all ℓ > ℓ2 it holds that Thitting(ℓ) exp β ρ

5 Application to Threshold Technologies We analyze a threshold technology, modelling scenarios in which the system is either highly valuable if the overall contribution of the players reaches a certain threshold τ, and less valuable otherwise:

V (ℓ) = Vlow ℓ< τ Vhigh ℓ τ Vlow < Vhigh , (18)

where ℓis the number of actively participating players, i.e., ℓ= P i ai and ai {0, 1}. The corresponding token values according to (1) are thus tlow = Vlow/Ttot and thigh =

Vhigh/Ttot. We are interested in the probability that the underline dynamics selects the high value (optimal) outcome,

phigh(ρ) := Pr a πρ[V (a) = Vhigh] . (19)

5.1 Stochastic stability (β regime) Theorem 6. For any threshold technology (18) with airdrop rewards (2) and vanishing noise (β ), the probability of selecting the high value outcome (19) undergoes a sharp transition given by the rewards ρ:

lim β phigh(ρ) = 1 ρ > ρc 0 ρ < ρc , ρc := α n τ Vhigh Vlow . (20)

For the edge case where ρ = ρc, the probability satisfies limβ phigh(ρ) = 1/ 1 + n τ . An immediate corollary of the previous result follows. Intuitively, the corollary states that there exists three regions: (i) for very high cost, the probability of selecting the good outcome vanishes no matter how we set the rewards, and thus the optimal strategy of the designer is to set no airdrop, (ii) for intermediate costs, though it is possible to set ρ > 0 such that the probability of selecting the good outcome tends to one, the designer still prefer to set ρ = 0, and (iii) for low costs there is ρ > 0 maximizing the designer s profit and making the probability of selecting the good outcome going to one. Corollary 2. For any threshold technology (18) with airdrop rewards (2) and vanishing noise (β ), the probability of selecting the high value outcome (19) is as follows: 1. For α n τ > Vhigh Vlow the probability of selecting the high value outcome vanishes for any airdrop reward ρ. Hence, and the best strategy for the designer is to give no airdrop rewards, which guarantees a profit of Vlow d V . 2. For α n τ < Vhigh Vlow the probability of selecting the high value outcome tends to 1 for any airdrop reward ρ > ρc. The optimal strategy (profit maximizing) for the designer is as follows: (a) For α n τ (Vhigh Vlow) (1 Vlow/Vhigh) it is still optimal for the designer to give no rewards (causing the probability of selecting the good outcome to vanish). (b) For α n τ < (Vhigh Vlow) (1 Vlow/Vhigh) the best strategy for the designer is to set airdrop rewards slightly above ρc < 1, which guarantees a profit of (1 ρc ϵ)Vhigh d V for any small ϵ > 0. Note that the intermediate regime in part 2a of the corollary above occurs only for Vlow > 0. Here high rewards could make the system succeed, but they are not optimal for the designer. For Vlow = 0 we have a single transition (either provide no airdrop or set the airdrop slightly above ρc).

5.2 Non-vanishing noise (β finite regime) In this section, we analyze logit dynamics for threshold technologies in the case of non-vanishing inverse noise β > 0. Research suggest that in practice people respond according to some specific value of β which is approximately the same across different games and situations they face (details in

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Figure 1: On the left, larger costs α increase the hitting time (100 repetitions 95% confidence). On the right, larger rewards values ρ help to maintain the dynamics above the threshold once it is reached.

[Georganas et al., 2025]). The next result provides useful bounds on the probability that the high value outcome is selected at equilibrium by the dynamics. Theorem 7. For any threshold technology (18) with airdrop rewards (2) and any inverse noise parameter β > 0, the probability of selecting the high value outcome (19) is monotone increasing in the rewards ρ and, in particular, it has the following form:

phigh(ρ) = 1 1 + C exp( ρB), B = β

n (Vhigh Vlow) ,

where C = C(αβ, n, τ) = 1 phigh(0)

phigh(0) does not depend on rewards ρ nor on the values Vlow and Vhigh. Based on the result above, we are able to characterize the optimal airdrop rewards for the designer. Theorem 8. For any threshold technology (18) with Vlow = 0, and with airdrop rewards (2) and any inverse noise parameter β > 0, the designer s profit (13) is

profit(ρ) = Vhigh 1 ρ 1 + C exp( ρ B) d V , (21)

where quantities C and B = β

n Vhigh are defined as in Theorem 7. Moreover the following holds: 1. For n β Vhigh the optimal strategy (profit maximizing) for the designer is to give no airdrop rewards, which guarantees a profit of Vhigh phigh(0) d V . 2. For n < β Vhigh the optimal strategy (profit maximizing) for the designer is to set an airdrop reward ρ ρ := 1 1/B = 1 n βVhigh . The probability of selecting the high value outcome for the designer s optimal rewards ρ is bounded as follows: phigh(ρ ) phigh( ρ) = 1 1+C exp(1 B) = 1 1+C exp(1 β Vhigh/n).

3. For n < β Vhigh (1 phigh(0)) the optimal strategy (profit maximizing) for the designer is to give strictly positive airdrop rewards.

5.3 Convergence time In this section, we study the time for logit dynamics to converge to its equilibrium (stationary distribution) and to the good outcome of the threshold function (ℓ τ). Specifically, we provide tight bounds on the mixing time (Theorem 9) and on the hitting time of a target value (Theorem 10).

Some intuition first. We observe experimentally (Figure 1) that lower costs α accelerates convergence to the desired high value region, while increasing rewards ρ helps to maintain the desired equilibrium (but it does not accelerate convergence). Intuitively, the dynamics converge quickly to an average contribution level ℓ which depends only on αβ:

ℓ = n pαβ , pαβ := 1 1 + exp(αβ) . (22)

Then, convergence to the desired high value region ℓ τ is fast for τ ℓ but becomes slow for τ > ℓ . This suggests that the convergence time grows with the gap τ ℓ and the hard case is when ℓ τ.

Formal analysis As for the mixing time, we leverage on the bounds for birth and death chains (Theorem 4). To this end, we note that in this particular case of threshold functions, the birth and death probabilities (15) assume a special form. This leads to the next theorem, which provides a lower bound on the mixing time for the useful scenario, that is, when the success probability is larger than the failure probability.

Theorem 9 (mixing time). For any threshold technology (18) and airdrop rewards (2), if phigh(ρ) > 1/2, then the mixing time can be bounded as follows: Tmix = Θ(Tcutoff) and

ˆπ([0, ℓ]) ˆπ(ℓ)p(ℓ) exp(αβ) exp(τ 1) n τ 1 . (23)

We next provide bounds on the hitting time of the good value, that is, to reach a contribution level ℓ= τ. The first part of the next theorem says that the dynamics converge quickly to a contribution level ℓ given by (22).

Theorem 10 (hitting time). For any threshold technology (18) and airdrop rewards (2), let Thitting(ℓ) be the expected time for the logit dynamics to reach state ℓstarting from state 0. Then, for ℓ defined as in (22), the following holds:

1. (Upper Bound). Thitting(ℓ ) O n2 ℓ n ℓ .

2. (Lower Bound). Thitting(τ) exp(αβ) ℓ+1

n ℓ τ ℓ , for all 0 ℓ τ. This implies, Thitting(τ) (1 + 1/ℓ )τ ℓ 1.

The second part of the theorem states that the time to reach the high value region increases with larger αβ, growing exponentially with the gap between ℓ and τ > ℓ .

6 Conclusions and Future Work

This paper presents a game-theoretic framework to address the dynamics involved in launching a new blockchain, specifically focusing on how contributions can be incentivized through token rewards and the possibility of support from an established mainchain. The analysis provided, offers both theoretical and practical insights into the design of blockchain launches, airdrop mechanisms, and the use of mainchain resources to achieve successful outcomes.

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