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Table of Contents
1 Basic of cryptographic payments 3
1.1 Preliminaries on cryptographic signatures 3
1.1.1 Binary and Hexadecimal 3
1.1.2 Hashing function. 4
1.1.3 Cryptographic Signatures 6
1.1.4 Takeaway 8
1.2 Blockchain Protocols 9
1.2.1 Consensus 9
1.2.2 Proof of Work 12
1.2.3 Block Difficulty adjustment 15
1.2.4 Longest-chain / Heaviest-chain rule 15
1.2.5 Halving 16
1.2.6 Blocksize 17
1.2.7 Why is it called “Proof of Work”? 18
1.2.8 Rise of ASICs 18
1.2.9 Layer 2 networks 19
1.2.10 Cursory analysis 19
1.2.11 Proof of Stake 21
1.2.12 Nodes and miners 21
1.2.13 Soft forks and hard forks 22
1.3 What is Bitcoin and why does it work? 23
1.4 More on mining 24
1.4.1 UTXO 24
1.4.2 Common assumptions in mining games 27
2 Probability spaces 30
2.0.1 Countably and Uncountably infinite 31
2.1 Partitions 32
2.1.1 Conditional Probability 33
2.2 Random Variables 34
2.2.1 Expectation 34
2.2.2 Conditional expectation 36
2.3 Bernoulli Processes and Bernoulli Schemes 38
2.3.2 Probability-preserving isomorphisms and comparison of expectations 42
2.4 Win-Lose-Pay Game 45
2.5 Probability distribution functions 46
2.5.1 Jensen’s inequality. 47
2.6 Poisson processes 47
2.6.1 Naturally occurring chain splits 49
3 Game theory basics 52
3.1 Single round games 52
3.1.1 Payoff Tables 53
3.1.2 Preferences and utility function 54
3.1.3 Matrix Games 55
3.1.4 N-players games and more terminology 56
3.1.5 Dominant Strategies 57
3.1.6 Examples 58
3.1.7 Nash equilibria 59
3.1.8 Examples 60
3.1.9 Determining Nash Equilibria by looking at a table. 60
3.2 Mixed strategies 62
3.2.1 Mixed strategies in real life 62
3.3.1 State of a game 63
3.3.2 Game Trees 65
3.4 Schelling points 67
4 Monopolizing Pool 69
4.1 Rough approach: Bernoulli’s Law of Large Numbers and Hoeffding’s bound 70
4.2 Binomial distribution and De Moivre-Laplace theorem 73
5 Basic double-spend game 76
5.1 Simplest Strategy: Charlie elects to mine his own chain when less than k blocks behind 78
5.1.2 A method for computing winning probability 80
5.1.3 Another method: difference equations 82
5.1.4 Computing expected values 84
5.2 Longer duration attacks 90
5.3 Stealth 6 block double-spend. 92
5.3.1 Negative binomial distribution 93
6 Censorship attacks 97
6.1 Worked example: two rational pools 99
6.1.1 Consideration for this choice of function. 101
6.1.2 There are many strategies 102
6.1.3 Case 1: (Non-compliant, Non-compliant) 103
6.1.4 Case 2 (Non-compliant, Mildly Compliant). Pool 1 is non-compliant, Pool 2 is mildly compliant. 106
6.1.5 Case 3 Both Pools Mildly compliant 113
6.1.6 Analysis: Low fee regime 114
6.1.7 Higher fee regime 117
6.1.8 Conclusion 117
7 Economics of Mining 119
7.1 Hashing as a market 119
7.1.1 Cournot oligopoly Model 119
7.1.2 A model for the mining market 121
7.1.3 Linear cost model 121
7.1.4 Monopolist pricing and collusion 124
7.1.5 Perfectly competitive markets 124
7.1.6 Nash Equilibrium in the mining market. 126
7.2 Market for ASICs 127
7.2.1 Four Regimes 128
8 Selfish mining 130
8.1 Exploiting the difficulty adjustment 131
8.2 The basic Selfish Mine 132
8.3 Market distortions 137
8.4 Markov Chains 138
8.5 Selfish mining analyzed via a Markov process with cuts. 140
8.6 Selfish mining with partial cooperation 143
8.7 Defense against selfish mining 144
9 Strategic Mining 146
9.1 Petty Compliant strategy and fee-undercutting 147
9.1.1 Fee-Undercutting 147
9.1.2 Fee-sniping vs. fee-undercutting 149
9.1.3 Emergence of petty compliant Miners 149
9.1.4 Selfish mining as incentive to use petty compliant mining 150
9.1.5 Measures against undercutting 151
9.2 Other Deviant mining strategies 152
9.2.1 Accepting bribes. 152
9.2.2 Block withholding 159
9.2.3 Protecting one’s own interests 160
9.2.4 Mine-from-behind 161
9.2.5 Extended example 161
9.2.6 Deferring to the alpha dog 165
9.2.7 Responding to exogenous demand for hashrate 168
9.2.8 Harmonic Mining / Switch Mining 170
9.2.9 Merge-mining, layer 2, sidechains and strategies playing out on other blockchains 172
9.3 Will there be a transition from default compliant to a fully strategic regime? 173
9.3.1 Block optimization and transaction selection 174
9.3.2 Overpaying pools 175
9.3.3 Direct signaling and optimizer-training 176
9.3.4 Purge pooling 176
9.3.5 Precedent: MEV and selfing mining in Ethereum 177
10 What discourages strategic mining? 179
10.1 Market Fragility Hypothesis 179
10.2 Repeated games and institutional norms. 180
10.2.1 Mining that can be punished 182
10.2.2 Problems with punishment 182
11 Declining Block Subsidy 184
11.1 Security budget 185
11.2 The mining gap 187
11.3 Problems with a mining gap 189
11.4 Undercutting 190
11.4.1 Private transaction brokering 192
11.5 Summary 193
12 Flattening ASIC cost curve 195
12.1 Hash deployment ratio 195
12.2 Purge Pools 196
12.2.1 Protection against a purge attack when HDR is low 197
12.3 The dynamical system determined by the difficulty adjustment 198
12.3.1 Phase transition to Galloping Gertie 205
12.4 Financialization of hashrate markets 207
13 Attacks 209
13.1 Why have there been no attacks yet? 210
13.2 Nation state / Authoritarian attacks 211
13.2.1 The empty-block attack 211
13.2.2 Obvious criticism of the empty-block attack 212
13.2.3 Layer 2 s under an empty-block attack 213
13.2.4 The seesaw attack 213
13.2.5 bribe-flooding attack 214
13.2.6 Empty-block purge attack 215
13.3 Checkpointing against a persistent 51 % attack 215
13.4 Buyout Attacks 218
13.5 Nuclear option: new proof-of-work algorithm 219
13.6 Less-than-brute-force attacks 221
13.6.1 Nudging towards a strategic regime 221
13.6.2 FUD and social attacks 222
13.6.3 Regulations 223
13.7 Goldfinger attacks 223
13.7.1 Slippery-slope attacks 225
13.7.2 Selfish mining as a slippery-slope attack 226
13.7.3 Low probability attacks 227
13.8.1 Additional security layers 232
13.8.2 Example: Purge Exploit and Buy Beware response 234
13.8.3 The government to the rescue 235
13.8.4 Economic majority vs. mining majority 236
13.8.5 A tetralemma 237
14 Direct Frontal Takeovers 239
14.1 Institutional response 241
14.2 Creeping corporate takeover 242
14.3 Alliances are beneficial 242
14.4 ESG takeovers 243
14.5 Government KYC takeovers 244
14.6 Overexposed bagholder exploit 244
14.7 Fee market and the long term viability 245
14.8 How much is censorship resistance worth? 246
14.9 67 % Fiat Attack 247
14.10 Permissioned Bypass 248
15 Nash bargaining 249
15.1 The Bargaining problem 250
15.1.1 Motivating Real World Problems 251
15.1.2 Motivating Game Theory Games 251
15.1.3 Dividing a surplus and the Nash bargaining solution 252
15.1.4 Exogenous probability of breakdown 254
15.1.5 What if players can determine the probability of breakdown? 256
15.2 Creating commitment 258
15.2.1 Enter blockchains? 259
15.2.2 Commitment Can Backfire 260
15.3 Mixed Strategies and Schelling Threats 260
15.3.1 Jane’s example, again 261
15.4 Extortion and griefing 262
15.4.1 Schelling attacks as an inexpensive way to destroy a network. 265
15.5 Wars of attrition 266
16 Coalitional game theory 267
16.0.1 Motivating examples 267
16.0.3 Standard coalitional game theory and the Bitcoin mining game 271
16.0.4 Solution concepts 272
16.0.5 Analysis of solution concepts for the weighted majority games and CBMG 279
16.0.6 The 3 player case 284
16.0.7 Discussion 288
17 Stock, Flow and Economic considerations 290
17.1 Double-spend exploits 293
17.1.1 Finality 295
17.2 Sabotage (Goldfinger) Attack 298
17.3 Determining the stock value of ASICs. 299
17.3.1 Moore’s law 300
17.3.2 flow-based miner cost model 301
17.3.3 Stock value of latent hashrate as a threat 302
17.4 Stable regimes 303
17.5 Efficiency, Security and Decentralization. 304
17.5.1 Efficiency 305
17.6 Theoretical pushout and epsilon-attacks 308
18 Bitcoin as it matures 310
18.1 Hyperbitcoinization 310
18.1.1 Decentralized Hyperbitcoinization 311
18.1.2 Centralized Hyperbitcoinization 314
18.1.3 Sovereign currency will not die 317
18.2 Will nation-states support Bitcoin? 320
18.2.1 Indifference to centralization and the Soft-Forkability hypothesis. 322
18.2.2 Bitcoin mining 323
18.3 Non-profitable mining 324
18.3.1 Will corporations protect their bags? 327
18.4 Geopolitical considerations in adversarial conditions 327
18.4.1 Nice and naughty nations. 331
18.4.2 The free-rider problem with a finite supply currency 333
18.5 Can the supply be increased? 334
18.6 Limits as a decentralized store-of-value 335
18.6.1 Asymmetric risk premium 336
18.6.2 Failure as a functional store of value 337
18.7 Beasts in the Jungle: Centralization, Capture and Collapse 338
18.7.1 Centralization 338
18.7.2 Capture 340
18.7.3 Collapse 342