Finite Blockchain Games

This paper studies the dynamic construction of a blockchain by competitive miners. In contrast to the literature, we assume a finite time horizon. Moreover, miners are rewarded for blocks that eventually become part of the longest chain. It is shown that popular mining strategies such as adherence to conservative mining or to the longest-chain rule constitute pure-strategy Nash equilibria. However, these equilibria are not subgame perfect.


Introduction
Since the introduction of the bitcoin consensus protocol by Nakamoto (2009 interesting conclusions. Here, we will try a related, but more elementary 27 analysis. 28 Speci…cally, in this paper, we model the construction of a blockchain as an 29 extensive-form game with …nite time horizon T . In each stage, the population 30 of n miners (or mining pools) strives to append the respective next block to 31 the existing blockchain. Thus, starting from the so-called genesis block, the 32 blockchain develops in a stochastic manner. Miners are assumed to earn 33 one token for any block that is contained in the longest chain at the end of 34 the game. 1 Now, being able to choose a parent block at libitum, miners may 35 intentionally try to create forks. A conservative miner always appends any 36 new block to the original chain, i.e., to the chain that contains the …rst child 37 block, thereof the …rst child block, and so on. We also consider the class of 38 mining strategies that follow the longest-chain rule, i.e., that append any 39 new block to one of the longest chains in the blockchain. We con…rm that 40 conservative mining and, in fact, any combination of strategies consistent 41 1 Should there be more than one longest chain at the end of the game, one such chain is chosen randomly. with the longest-chain rule, form Pareto e¢ cient Nash equilibria. However, 42 we also show that, under the assumptions made below, these equilibria are 43 not subgame perfect (Selten, 1965). This contrasts with …ndings of the recent 44 literature that has found such strategies to be consistent even with the more 45 restrictive concept of Markov perfect equilibrium.

46
The rest of the paper is organized as follows. Section 2 recalls the formal de…nition of a blockchain. Section 3 introduces …nite blockchain games. We   Thus, a blockchain B consists of (T + 1) blocks, where T is the time horizon.

59
The block b 0 is referred to as the genesis block. Any two blocks may be 60 related to each other by a parent-child relationship. Finally, each block except 61 the genesis block has a miner assigned to it. An example of a blockchain is shown in Figure 1. The numbers close to the circles are the respective miner assignments. 66 We will impose the following two additional requirements: 67 (a) each block except the genesis block b 0 has precisely one parent, i.e., for 68 any t 0 > 0, there is precisely one t such that b t W b t 0 69 (b) the parent has a lower index than the child, i.e., b t W b t 0 implies t < t 0 .   Miners'payo¤s are determined as follows. After stage T , one of the longest 94 chains C in the blockchain B T is drawn with equal probability. Each miner 95 i 2 N receives one token for each block b 2 Cnfb 0 g assigned to her. Miners 96 are risk-neutral and maximize the expected number of tokens they receive.

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The stochastic game introduced above will be referred to as a …nite of one of the longest chains. Note that the longest-chain rule is a class of 108 strategies, rather than a single strategy. 109 We start by studying Nash equilibrium (Nash, 1950). The following result 110 says that conservative mining, and likewise following the longest-chain rule, 111 constitute Nash equilibria in pure strategies.  Proof. (Conservative mining) Suppose that all miners j 2 N nfig are con-116 servative. We have to show that miner i has no strict incentive to deviate 117 from conservative mining. Assume …rst that i adheres to the candidate equi-118 librium strategy. Then, the blockchain develops into a single chain consisting 119 of (T + 1) blocks, and miner i receives one token for each block she mined. In this section, it will be shown using two examples that the considered Nash 130 equilibria need not constitute a subgame-perfect equilibrium (Selten, 1965). 131 We begin with the conservative mining equilibrium.  However, it is optimal here for miner 1 to work on b 2 because this allows her, 142 with probability 1=2, to realize a token for the block b 2 .

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Thus, conservative mining is not subgame-perfect. But neither is the longest-144 chain rule, as the next example shows.