Modelling Bitcoin with Game Theory

Bitcoin is the world’s first and most popular decentralized cryptocurrency with a market capitalization of over 650 billion dollars.

Bitcoin is a chef-d’oeuvre, a masterpiece of different disciplines. Cryptography, economics, mathematics, game theory and even philosophy!

The users of Bitcoin are called clients.

Clients have accounts called addresses.

A client can send bitcoins to another client in a transaction committing it to the global, add-only ledger, the blockchain.

The blockchain is maintained by a network of miners who are rewarded for their efforts in bitcoins.

Miners are in charge of recording the transactions in the blockchain, which determines the ownership of Bitcoins.

This article focuses on how bitcoin is an excellent application of game theory.

What is game theory?

Game theory is the discipline that explores mathematical games, the interaction between players, and their decisions as influenced by incentives or payoffs following strict rules.

If you are a player in such a game, you must consider the choices of other players (and assume they are taking yours) when contemplating your strategy.

A mathematical game is any set of situations whose result is dependent on the actions of two or more players.

A player is a strategic decision-maker in a game. In certain situations, players have to make decisions called actions.

A strategy is a plan that tells the player what move to choose in every possible position.

Payoffs are the rewards gotten from different possible outcomes in any form.

What does this have to do with bitcoin?

The bitcoin blockchain is a system where everyone’s transaction history is a chain of blocks with dependencies, with miners computing hashes competing to propose the next valid block.

The bitcoin protocol uses the Proof of Work consensus algorithm to agree on the correct version of the ledger within the Bitcoin network.

If any miner controls more than 50 per cent of the network’s CPU power, they could temporarily modify the order of transactions or even double-spend some of its coins. This would make the system no longer be censorship-resistant and reversion-resistant.

To encourage miners to behave honestly, Bitcoin provides an incentive mechanism to ensure their self-interest is linked to the network functioning well.

To analyse the incentives of bitcoin, we model the protocol as a game.

First, who are the players?

The players in this protocol are the miners. For this article, I explain the game with 3 imaginary players Dele, Ibukun, and Ufuoma — my top 3 subscribers.

What are the payoffs?

Payoffs are calculated by subtracting costs from rewards.

Costs

In the bitcoin protocol, miners have to generate hashes to make blocks. By “generate”, I mean guess.

Solving the maths problem in Proof of Work depends on how fast a miner can generate hashes, increasing their probability of guessing the right numbers. The miner with the highest “hash rate” has a greater chance of being the first to get the solution to the problem. A higher hash rate is only possible with special mining hardware, and a lot of electricity — both very expensive.

The costs of mining bitcoin are the fixed cost of mining hardware and the variable cost of electricity.

Rewards

The block reward, currently 6.25 BTC.

Miners are rewarded with the transaction fees of transactions included in their blocks.

How are the miners’ interactions timed?

All blocks are mined at the same time.

What choices do they have to make?

Every miner has a transaction pool. They get to choose what transactions to include in their block. A miner could choose to include transactions with the highest transaction fees, only transactions directly paying them, or even mine an empty block!

However, since miners powering up the Bitcoin network also receive a cut on the transactions included in a block, it would be counterintuitive to choose to mine an empty block since you would make no money from transaction fees.

Miners also have to choose what blocks to begin mining on. Mining on the honest chain is honest behaviour. The longest chain is the version of the blockchain accepted as the valid version of the blockchain by all nodes in the bitcoin network. It is the chain of blocks that took the most effort to build.

Miners are incentivized to add their blocks to the longest chain because only blocks mined on the longest chain are eligible for block rewards. It would be very wasteful to use a lot of resources to compute proof of work and then to add a block to a chain that is ineligible for block rewards.

Finally, the only way an attacker can compromise or tamper with the Bitcoin blockchain is by gathering at least 51 per cent of the hashing power in a 51% attack.

To carry out a 51 per cent attack, an attacker would need to gather a lot of special mining equipment. If the attacker manages to do so, they would then have to compromise the network. However, the attacker would not be incentivized to do so since they have invested a lot of money in mining hardware.

If they manage to attack the network, the cryptocurrency would immediately lose significant value as one of its properties is that its blockchain is immutable. This would cause the attackers to have wasted a lot of money.

For maximum payoffs, it is best to follow the rules of the network.