Networking as a decentralization bottleneck

We look at transaction propagation in the absence of a properly incentivized networkinglayer.

Iragedy of the commons. Carrett Hlardin explained the tragedy of the commons in his essay 4in the following way: "Picture a pasture open to all. It is expected that each herdsman will try tokeep as many cattle as possible on this commons. ... Therein is the tragedy. Each man is lockedinto a system that compels him to increase his herd without limit, in a world that is limited, Ruinis the destination toward which all men rush, each pursuing his own best interest in a society thatbelieves in the freedom of the commons."

Relayer bandwidth is a common resource.Computation, storage and bandwidth are commonresources used in blockchains. While computation has an incentiveized marketplace today, noneexist for bandwidth and storage". As a result, blockchains today rely on the altruism of minersand full nodes to relay blocks and transactions.

In the case of blocks, miners have an incentive to relay their blocks and receive the latest blocks inorder to prevent the blocks they mine from being orphaned%. However, they have little incentive toproactively propagate blocks mined by the other miners. They are in fact incentivised to withholdthe block and propagate it only after they have mined a block on top of it. This not only savesthem from bandwidth costs but also gives them an edge with respect to the other miners.

Free-rider problem. A market failure that occurs when individuals prefer to let others make theinvestment and yet have access to the benefits of the investment.

Iransaction propagation is not incentivized.Transaction propagation is susceptible to theree-rider problem where nodes would like to save bandwidth costs by not propagating transactionsand rely on others to do it.

Analysis. Consider a set of n full nodes. Each node i chooses a strategy of forwarding transactions or not, represented by s; E S, = (0.11. Let S = S, be the set of all possiblestrategy profiles. Each node has a payoff function f (s) for every strategy profile s E S. Letf(s) = (f1(s), /2(s),..., /(s)). Given a game (S, f) as delined among n full-nodes, we attempt toanalyze the Nash equilibrium, the condition under which it exists and its implications.

Let s-i be the strategy profile of all players except player i, i.e. s = (Si,s-). Let the utility thatnode gets out of the blockchain platform be represented by u;(s). A node choosing to forwardtransactions can only bring more utility to the blockchain platform, i.e.

Let the cost that a node incurs if it chooses to forward transactions be c. Assuming no incentiveto forward transactions, the payoff for nodes can be written as:

A Nash equilibrium is said to exist at a strategy profile if the player gains nothing by changing hisstrategy unilaterally while other players stick to their existing strategy.