1. Blockchain Mining Games
Kiayias, Aggelos, et al. "Blockchain mining games." Proceedings of the 2016 ACM Conference on Economics and Computation. ACM, 2016.
Presented by Zhenpeng Shi
7/19/2019

2. Introduction Selfish Mining
Miners withhold releasing mined blocks in order to get larger revenue than their fair share.
Simulation results shows that to avoid selfish mining, at least 67% miners must be “honest” miners.
Can we prove it theoretically?
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Fig. 1. Mining power α above which selfish mining trumps honest mining, function of the propagation factor γ.[1]
[1] Eyal, Ittay, and Emin Gün Sirer. "Majority is not enough: Bitcoin mining is vulnerable." Communications of the ACM61.7 (2018):

3. Introduction Blockchain mining games
Game theory: the study of mathematical models of strategic interaction between rational decision-makers.
Nash equilibrium: a state where no player has anything to gain by changing only its own strategy.
What we want to see: at Nash equilibrium, every miner gets its fair share of reward, which is proportional to its relative computational power.
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4. Introduction Blockchain mining games
Each miner is a player in the game, its object is to maximize reward.
Miners are selfish, which yields a non-cooperative game.
Miners’ strategies are:
(1) which block to mine (what to compute next)
(2) when to release mined blocks (when to release the results of computation)
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5. Model Two complete-information game models (1) immediate-release game
Every miner releases immediately the mined blocks
Can show that the threshold is 0.361≤ ℎ 0 ≤0.455
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6. Model Two complete-information game models (2) strategic-release game
Miners can withhold releasing blocks.
To ensure complete information, add the restriction: miners must immediately announce a successful mining of a block, but not the block it self.
Can show the threshold is bounded by ℎ 0 ≥0.308
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7. Model Parameters 𝑛 – the number of miners (players)
𝑝=( 𝑝 1 ,…, 𝑝 𝑛 ) – the probability that miners succeed in solving the crypto- puzzle, which reflects their relative computational power; sum up to 1: 𝑖=1 𝑛 𝑝 𝑖 =1
𝑑 – the depth of the game, the payment for mining a new block is paid after 𝑑 new blocks attached to it
𝑟 ∗ – reward for mining a block; assume 𝑟 ∗ is constant and normalized to 1
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8. Model
Definitions
Public state: a rooted tree. Every node is labeled by one of the players. The nodes represent mined blocks and the label indicates the player who mined the block. Every level of the tree has at most one node labeled 𝑖.
Private state: similar to the public state except it may contain more nodes called private nodes and labeled by 𝑖. The public tree is a subtree of the private tree and has the same root.
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9. Model
Definitions
Strategy: A pure strategy of player 𝑖 consists of two parts:
Mining: to select a node of the current public state to mine
Releasing: a (perhaps empty) private part of the player’s state which is added to the public state
Frontier strategy: to release any mined block immediately and select to mine one of the deepest nodes.
In a fair blockchain mining game, frontier strategy is expected to be played by all players.
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10. Model
Definitions
Payments: For some nodes of the tree, the miners who discovered them will get a fixed payment (normalized to 1). The payments comply with the following rules:
(1) the nodes that receive payment must form a path from the root, which implies that at every level of the tree exactly one node receives payment.
(2) among the nodes of a single level that satisfy the above path restriction, the first one which succeeds in having a descendant 𝑑 generations later receives payment.
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11. The Immediate-Release Game
Assume that all miners who follow the Frontier strategy act as a single miner.
2-player game: Miner 1 is the miner whose optimal strategy we wish to determine and has relative computational power 𝑝; Miner 2 is assumed to follow the Frontier strategy and have collective relative computational power 1−𝑝.
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12. The Immediate-Release Game
Prune away abandoned branches.
The state will be a long path followed by two branches, one for each miner of lengths 𝑎 and 𝑏.
Denote the state by (𝑎,𝑏), notice that 0≤𝑎≤𝑏+1.
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13. The Immediate-Release Game
Mining states M: both miner keep mining their own branch.
Capitulation states C: Miner 1 gives up its branch and continue from a block of the other branch.
Winning states W: Miner 2 gives up its branch and continue from the newest block of other branch.
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M-orange, C-green, W-red

14. The Immediate-Release Game
Let 𝑔 𝑘 𝑎,𝑏 denote the expected gain of Miner 1 when the frontier advances by 𝑘 new levels starting from an initial tree at state (𝑎,𝑏)
Constant 𝑔 ∗ is the expected gain per level in the long run.
Define:
Potential function 𝜑(𝑎,𝑏) denotes the advantage of Miner 1 for currently being in state (𝑎,𝑏).
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15. The Immediate-Release Game
Consider three cases:
(1) 𝑎,𝑏 ∈𝑀. Both miners continue mining, with Prob. 𝑝, the new state is (𝑎+ 1,𝑏), with Prob. 1−𝑝, the new state is (𝑎,𝑏+1)
(2) 𝑎,𝑏 ∈𝐶. Miner 1 abandons its branch and the new state is (0,𝑠), where 0≤𝑠<𝑏
(3) 𝑎,𝑏 ∈𝑊. Miner 2 abandons its branch and the new state is (0,0), Miner 1 gains 𝑎
By definition, 𝑔 0 𝑎,𝑏 =0.
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16. The Immediate-Release Game
Get a similar recurrence for 𝜑
Fix 𝜑 0,0 =0. Notice that potential function 𝜑(𝑎,𝑏) is always non-negative.
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17. The Immediate-Release Game
Frontier is a Nash equilibrium if, having fixed the strategy of all miners except 𝑖 to Frontier, the best response of a miner 𝑖 has expected gain per level equal to 𝑝 𝑖 (the probability that miner 𝑖 succeeds in mining a new block, or its relative computational power), i.e., 𝑔 ∗ = 𝑝 𝑖 .
Theorem: In the immediate-release model, Frontier is a Nash equilibrium when every miner 𝑖 has relative computational power 𝑝 𝑖 ≤0.361.
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18. The Immediate-Release Game
Important lemmas in proof:
(1) Miner 1’s winning probability starting at state (𝑎,𝑏) is no more than 𝑝 1−𝑝 1+b−a .
(2) If Miner 1 capitulates from state (0,1), no other state except (0,0) will ever be reached.
(3) If 𝑝<0.361, then state 0,2 is not a mining state.
(4) For 𝑝<(3− 5 )/2, if (0,2) is not a mining state, then (0,1) is not a mining state.
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19. The Immediate-Release Game
Upper bound of 𝑝
Theorem: When Miner 2 plays Frontier, the best response strategy for Miner 1 is not Frontier when 𝑝≥0.455.
Idea of the proof:
Fix the depth 𝑑 (e.g. let 𝑑=3), then construct 𝜑(𝑎,𝑏) so that it’s a valid potential function of the game, from which we get a relationship between 𝑔 ∗ and 𝑝.
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20. The Strategic-Release Game
Similar to the immediate-release game.
𝑎 𝑟 ≤𝑎 denotes the number of released blocks on Miner 1’s branch.
All blocks on Miner 2’s branch are always released.
It can be 𝑎>𝑏+1.
Assume that
(1) if 𝑎≤𝑏, then 𝑎 𝑟 =𝑎;
(2) if 𝑎>𝑏, then 𝑎 𝑟 =𝑏,
that is, 𝑎 𝑟 =min⁡{𝑎,𝑏}
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21. The Strategic-Release Game
Define the expected gain of Miner 1 as
Consider the following two cases:
(1) 𝑎≤𝑏. Miner 1 can choose to capitulate or to mine, which is the same as in the immediate-release game.
(2) 𝑎>𝑏. Miner 1 has another option to release an additional block and lead the game to state (𝑎−𝑏−1,0), in which case Miner 2 will capitulate.
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22. The Strategic-Release Game
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Theorem: In the strategic-release model, Frontier is a Nash equilibrium when every miner 𝑖 has relative computational power 𝑝𝑖≤ 0.308

23. The Strategic-Release Game
Idea of the proof:
For states 𝑏+1,𝑏 , Miner 1 probably will release all blocks.
For states (𝑎,𝑏) with 𝑎>𝑏+1, Miner 1 is safe not to release blocks until Miner 2 catches up within distance one.
Define a potential function 𝜑 (𝑎,𝑏) which tends to overestimate the optimal potential.
Prove that when 𝑔 ∗ =𝑝 and 𝑝≤0.308, 𝜑 (𝑎,𝑏) satisfies the recurrence of 𝜑 (𝑎,𝑏)
The optimal gain per step cannot exceed 𝑝, that is
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24. Conclusion
Provide a game-theoretic model to analyze bitcoin miners’ behavior.
Identify the thresholds of the relative computational power below which Frontier is a Nash equilibrium.
For the immediate-release game, the threshold is 0.361≤ ℎ 0 ≤0.455
For the strategic-release game, the threshold is bounded by ℎ 0 ≥0.308
Future work:
Tighten the thresholds.
Consider the incomplete-information game.
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25. Thank You!