There is a rich body of literature in economics examining strategies that rely on committing sabotage against other participants in games (Chowdhury and Gürtler, 2015). Indeed, in certain games, such as contests with a fixed prize that is awarded to one of a finite number of players, strategies based on attempting to reduce the chances of an opponent in order to improve one’s own can be incentivized (Konrad, 2000). Typically, existing research on this question adopts an approach that involves attempting to incorporate the incentives of participants to harm others directly into the payoff matrices of the games considered.

A related idea, introduced by Buterin (2018b) in the context of analyzing blockchain systems, defines a “grief” as a strategy that causes harm to both the participant executing this strategy and other participants in a game. The word “grief” draws upon the slang term used in video game communities for styles of play that seek to harm others despite not being useful or indeed being harmful to the player carrying them out (Leonardos et al., 2023). Buterin additionally provides a measure of the effectiveness of griefs, which he calls the “griefing factor,” indicating the ratio of the harm that an attack does and its cost to the attacker (Buterin, 2018b).

To illustrate this concept, we consider a game with two players Alice and Bob whose payoffs are given in Table 1. It is a (weakly) dominant strategy for each player to play A. If Alice thus expects Bob to play A, she can nonetheless play B. Playing this strategy requires Alice to accept a cost: she loses the opportunity to earn $1. However, by accepting this cost to herself, Alice can inflict even greater economic harm upon Bob, as she prevents him from receiving $2. Hence, Alice has the ability to grief Bob with a griefing factor of 2.

Griefing factors provide a useful measure in that they allow for identification of the presence of particularly effective sabotage strategies in a game. As griefing factors can be calculated in terms of the payoffs of the game, one does not need to know in advance what rewards an attacker may receive for successful sabotage in order to perform an analysis of available griefing factors. Thus, when designing a game, a mechanism designer can attempt to ensure that all of the griefing strategies available in that game have griefing factors that are as small as possible, implying that whatever external incentives an attacker may have would need to be correspondingly large in order for it to be worthwhile to perform an attack.

In this article, we analyze the mathematical properties of griefing factors. While this analysis is general, our examples are largely drawn from the setting of blockchains. Indeed, this notion is perhaps particularly relevant in the context of blockchain systems due to their heavy reliance on mechanism design, in consensus algorithms (Nakamoto, 2008; Buterin, 2014) as well as in applications. We note several additional reasons that a griefing factor perspective is natural in the setting of blockchain systems.

• Interactions with different blockchain applications on a smart contract platform such as Ethereum (Buterin, 2014) can be linked so that a transaction that attempts to make two interactions either executes both successfully or has no effect, permitting novelties such as flash loans (Qin et al., 2021). Thus, attackers will often have interests that are not well captured by the internal incentives of the game designed by any given developer. In Ford and Böhme (2019), it is argued that, for these reasons, permissionless systems that are designed to be resistant to attacks by economically rational actors must actually be resistant to attacks by Byzantine actors in order to be secure. However, if one can show that the griefing factors of any actions available to an attacker are sufficiently small, then one can recover useful notions of economic security even in the presence of external incentives, as long as one can assume some upper bounds on the external rewards that the attacker can realistically obtain.

An extensive body of literature (Konrad, 2000; Chowdhury and Gürtler, 2015) considers games in which it is in the interests of players to cause harm to other participants. Further behavioral research (Abbink and Sadrieh, 2009) has shown that there are populations that do not require financial incentives in order to be incentivized to harm other players in economic games. This phenomenon is related to psychological studies on “trolls” (March 2019) who exert effort to antagonize others, particular in semi-anonymous Internet settings.

In defining griefing factors (Buterin, 2017; Buterin, 2018a; Buterin, 2018b), has provided a measure of a strategy’s effectiveness in doing harm to other participants relative to its cost that does not require a priori knowledge of what incentives participants may have to seek to do this harm. Buterin has discussed this concept principally in the context of analyzing blockchain settings. He notes in particular that griefs with griefing factors at least equal to one must exist in settings where there is “speaker–listener equivalence” (Buterin, 2017), and in Buterin (2018a), he considers the impact of a specific grief on economic equilibria of supply and demand for validators in a simplified model of a proof-of-stake system. Griefing factors have subsequently been used to analyze the effectiveness of specific attack strategies in various other blockchain systems, notably attacks on payment channels (Mazumdar et al., 2020; Mazumdar et al., 2022).

In Leonardos et al. (2023), griefing in a specific model of proof-of-work mining is related to a generalized notion of evolutionarily stable allocations explored in Schaffer (1988) in the context of games with a fixed reward. In that setting, it can be observed that participants are incentivized to act “spitefully” to drive competitors out of the competition. Indeed, Leonardos et al. (2023) observe that the unique Nash equilibrium in the mining game is not evolutionarily stable. Instead, evolutionarily stable allocations have a griefing factor of one with respect to the Nash equilibrium.

We expand on the mathematical framework allowing the study of notions of griefing and griefing factors. In Section 5 we illustrate these ideas by examining examples of griefs in several prominent blockchain applications. Notably, we provide a detailed analysis of various griefs between participants that are possible under the Ethereum consensus algorithm.

In Section 6.1 we explore questions on how the presence of players willing to engage in griefing affects the equilibria of a game. For any game, we define a collection of derived games in which the players obtain utility from griefing each other. As one varies the amount of utility that the different players derive from doing one unit of harm to other players (namely, the griefing factors at which the players are willing to engage in griefs), the equilibria in the derived games change. This can be represented geometrically through diagrams where each point in the diagram represents a given derived game in which the players are willing to grief at a given set of griefing factors. On this basis, one can consider the regions of the diagram where different equilibria are present and how the equilibria evolve as one moves along a path through the diagram. In Section 6.2 we present several results concerning the geometric structure of these regions. For example, we show in Proposition 4 that the boundaries of the region where a pure Nash equilibrium is present are defined by hyperplanes, although we see in Example 5 that the regions where mixed equilibria are present can have more complicated geometric structures. We consider examples illustrating how these results apply to blockchain applications. Notably, in Example 4 we extend results presented in Leonardos et al. (2023) to consider the equilibria that are present in the presence of griefers in a mining game that models proof-of-work systems.

We consider games of the form (N, A, u), where.

• N is a finite set of players of cardinality |N| = n;

We generally denote i’s action as x _i ∈ A _i , and we denote the tuple of actions of the other users by x _−i ∈ A _−i , consisting of an action x _j ∈ A _j for all j ≠ i. Sometimes we condense this by writing x = (x _i , x _−i ). We can compute the utility of the jth participant given these actions as u _j (x) = u _j (x _i , x _−i ) = u _j (x ₁, … , x _n ) for any j ∈ N.

We take Δ i = s i = s i x x ∈ A i ∈ 0,1 | A i | : ∑ x ∈ A i s i x = 1

as naturally identified with the set of mixed strategies that the participant i ∈ N can take; in particular, for s _i ∈ Δ _i and x ∈ A _i , we denote by s _i (x) the probability that i selects the alternative x while playing the mixed strategy s _i . Thus, Δ = ∏ i = 1 n Δ i corresponds to the set of tuples of a mixed strategy for each participant.

In Section 6 we consider utility functions that are modified to depend on the griefing factors at which participants are willing to perform griefs. Specifically, we will take a set of values (λ ₁, … , λ _n ) where λ _i measures the willingness of i to grief. We will also take a tuple of “base” utility functions u ₁, … , u _n , and we will present various formulations in each of which we will have a tuple of derived utility functions that can depend on u ₁, … , u _n as well as λ ₁, … , λ _n , as described below in Eqs 4, 5. These derived utility functions are chosen in such a way as to take into account utility that griefing players derive from causing harm to other players playing the game, defined by u ₁, … , u _n . Players in a given game are assumed to be economically rational in that they seek to maximize their utility as given by the relevant derived utility functions that include this utility from griefing. We will suppose that players take actions simultaneously and independently unless otherwise noted. Finally, we assume that these rationality, independence, and simultaneity assumptions, as well as the utility functions considered (including both the “base” utility functions u ₁, … , u _n and the various derived utility functions), are common knowledge.

We fix definitions of griefing factors that follow those of Leonardos et al. (2023), covering both scenarios in which an attacker griefs another individual participant and those in which an attacker griefs the broader set of other participants.

Definition 1.

Let i, j ∈ N . Take x = (x _i , x _−i ) ∈ A assigning an action to each player, namely, x _i ∈ A _i and x _−i ∈ A _−i . Suppose x i ′ ∈ A i is such that u i ( x ) = u i ( x i , x − i ) > u i ( x i ′ , x − i ) . Then we define the individual griefing factor of the action x i ′ against j with respect to x as: G F i , j x i ′ , x − i , x = u j x − u j x i ′ , x − i u i x − u i x i ′ , x − i (1) and the total griefing factor of the action x i ′ with respect to x as: G F i x i ′ , x − i , x = ∑ j ≠ i u j x − u j x i ′ , x − i u i x − u i x i ′ , x − i . (2) We can extend these concepts to the case where u i ( x ) = u i ( x i ′ , x − i ) , but u j ( x ) − u j ( x i ′ , x − i ) > 0 for some j or ∑ j ≠ i u j ( x ) − u j ( x i ′ , x − i ) > 0 , by adopting the convention that the corresponding griefing factors are infinite.

Remark 1.

Note that we assume u i ( x ) ≥ u i ( x i ′ , x − i ) in order for the griefing factor of ( x i ′ , x − i ) with respect to x to be defined. If we did not have this constraint, we would have G F i , j x i ′ , x − i , x i , x − i = G F i , j x i , x − i , x i ′ , x − i

for all x i ′ , x i ∈ A i and x_−i ∈ A_−i. Then, for example, situations in which deviation by i from x_i to x i ′ benefits both herself and j would result in the same griefing factors as those where i inflicts an equivalent amount of harm on herself in order to harm j, even though we think of these two situations as being qualitatively different. Thus, we simply avoid griefing factor values having such multiple possible interpretations by restricting the definition of griefing to situations where players would be accepting in-game harm to themselves. Indeed, we think of the attacker not receiving an economic benefit from an attack, in terms of her in-game incentives, as being an essential aspect of what it means to engage in a “grief”, and these situations are the focus of this work.

On the other hand, note that Definition 1 still allows for the possibility that u i ( x ) > u i ( x i ′ , x − i ) but u j ( x ) − u j ( x i ′ , x − i ) < 0 , which results in negative griefing factors. In most of the results of this work, we will limit ourselves to considering positive griefing factors; however, the perspective of negative griefing factors may be relevant to analysis of the behavior of a participant i who is altruistic.

We further illustrate the notion of griefing factors by calculating the griefing factors of several griefs on the Ethereum consensus protocol (Buterin, 2022).

Example 1.

We consider the economic games of the Ethereum Beacon Chain, as they are implemented after the Paris hard fork (“the merge”) (Buterin, 2021). In this game, validators perform several roles. Validators are randomly selected to produce new blocks, and additionally once per epoch (approximately 6.4 min) each validator is tasked with attesting to the most recent block in the preceding slot (a 1/32 division of an epoch) (Buterin et al., 2022; Bedawala and Salot, 2023). Also included in attestations are references to previous blocks, referred to as the “target” and the “source”. The information on target and source blocks that is provided in attestations is used in the process of determining when a block is finalized (see Buterin and Griffith, 2017). The system is calibrated so that if all actors, including block producers and attestors, perform their roles perfectly then each participant will earn on average a base reward of B per epoch (Buterin et al., 2022). This base reward is a function of the number of active validators, and as such it gradually changes over time as validators join or leave the network (Buterin, 2022).

In order for the attestation to be eligible for rewards, it must be “timely” (Buterin et al., 2022): that is, it must be provided in the next slot to obtain a reward for indicating the preceding block, in the next 32 slots to be rewarded for indicating the target, and in the next 5 slots to be rewarded for indicating the source. Attestors are rewarded B ⋅ 14 64 ⋅ m n for providing a timely attestation that correctly identifies the previous block, B ⋅ 26 64 ⋅ m n for providing a timely attestation that correctly identifies the target, and B ⋅ 14 64 ⋅ m n for providing a timely attestation that correctly identifies the source. Here n is the number of validators assigned to produce an attestation and m is the number of validators who provide timely attestations correctly identifying the previous block, target, or source, respectively. The correct previous block, target, and source are determined by the values that are ultimately finalized. If an attestor fails to include a timely target or a timely source, she is penalized B ⋅ 26 64 or B ⋅ 14 64 , respectively. However, failure to identify the previous block is exempt from penalties (Buterin et al., 2022). Block producers are rewarded B 8 ⋅ 14 54 for including a timely attestation that correctly identifies the previous block, B 8 ⋅ 26 54 for including a timely attestation that correctly identifies the target, and B 8 ⋅ 14 54 for including a timely attestation that correctly identifies the source. Block producers are not penalized for the non-inclusion of an attestation.

We analyze several griefs in this system. For all of these analyses, we assume that the result as to which blocks are finalized does not ultimately change. We express griefing factors in terms of B, m, and n. We also provide an approximate value of each griefing factor after the symbol “≈” based on current typical values for B, m, and n. As of June 2023, the current number of active validators is approximately 625000 (Bitfly, 2023), so we estimate that the number of participants in a given slot is n = 625000/32 = 19531. We also take m = 19336 so that the proportion of validators completing each task is m n = . 99 , corresponding to historic averages. Finally, the current value of B is roughly 14500 Gwei (Bitfly, 2023).

We determine griefing factors with respect to default strategies of honest participation. Specifically, in the notation of Definition 1, x_−i corresponds to the m attestors other than i providing correct, timely attestations and block producers other than i providing blocks that include all possible attestations. The action x_i corresponds to i also providing correct, timely information as an attestor or as a block producer when applicable, and x i ′ is the alternative action described in each grief.

• The attestor i griefs the block producer and the other attestors by publishing her attestation beyond the timeliness deadline for indicating the previous block. This costs the griefer 14 B 64 m + 1 m in lost attestation rewards. This deprives the block producer of a reward of B 8 ; however, a subsequent block producer receives a reward of B 8 ( 26 + 14 54 ) for other components of this attestation. Each of the m other attestors also has their reward reduced from 14 B 64 m + 1 m to 14 B 64 m n . Thus, this action by i has a griefing factor of

This can also be considered as a grief against the block producer j individually, noting that the block producer is also an attestor. G F i , j = B 8 + 14 B 64 m + 1 n − 14 B 64 m n 14 B 64 m + 1 n = 4 n + 7 7 m + 7 ≈ . 577 .

• A block producer i can grief the community of attestors (other than herself) by not including an attestation from j, hence causing it to fall beyond the one-slot period for timely reporting of the previous block. This reduces i’s block-producer reward by B 8 , and the reward for attesting to the previous block that i and each of the m − 1 other attestors receive is reduced from 14 B 64 m + 1 n to 14 B 64 m n . The attestor j is also deprived of a reward of 14 B 64 m + 1 n . The next block producer can include this attestation for rewards related to the target and the source, receiving B 8 ( 26 + 14 54 ) additional block-producer rewards. Thus, this action has a griefing factor of

As a grief against the censored attestor j individually, this gives a griefing factor of G F i , j = 14 B 64 m + 1 n B 8 + 14 B 64 m + 1 n − 14 B 64 m n = 7 m + 7 4 n + 7 ≈ 1.732 .

We see that the dependence on B can be canceled from the numerators and denominators of these expressions, so these griefing factors can be viewed as functions of m and n. Additionally, the two griefs against individuals are indistinguishable from the point of view of the consensus algorithm. Thus, they are an example of speaker–listener equivalence, as discussed in Buterin (2017); hence their griefing factors are reciprocals and, in particular, one of them must be at least one.

One can consider other griefs involving attestors not including correct, timely targets and sources. The griefing factors of these griefs are stated in Table 2 in terms of n and m and then using the approximate values of n and m indicated above. Note that under the current (Buterin et al., 2022) parameters for timeliness, failing to provide either a correct, timely target or source implies that the attestor also fails to provide a correct, timely previous block. Hence, the effects of failing to timely indicate the previous block are included in the griefing factors in the corresponding columns below. Additionally, note that block producers can only delay an attestation by one block, so it is not typically possible for them to prevent its inclusion within the time limits for timely targets and sources.

Another example of a grief on a cryptoeconomic system is considered in Leonardos et al. (2023) in the setting of proof-of-work mining games. We further consider this game in Example 4 using the tools that we develop in Section 6.1.

Note that it is also possible to have griefs that do harm that is not purely monetary. We present brief notes on cryptoeconomic systems in which griefers can expend resources to cause delays in the normal operation of the system.

Example 2 (Delay griefs).

• In Arbitrum, if contrasting states of the rollup ¹ are submitted by validators, then there is a process designed to prove that at least one of them is incorrect (Offchain Labs Developers, 2022). As the final states differ, there is guaranteed to be some specific step in the off-chain computation on which the two parties disagree. Via an interactive game where parties progressively commit to points of disagreement in their series of computations, such a step can be identified and verified on-chain. Attackers can place malicious stakes that require the system to use this process and then use the maximum amount of time they are allowed before responding to each step of the interactive game, causing delays. The cost of such an attack is considered in Offchain Labs Developers (2022).