What is the reasons for halving the rewards?
Why not, for example, let the reward be always the same and the amount of currency always growing? Are the reasons technical or just purely ideological?
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Because the block halving incentives getting into Bitcoin NOW, rather than later. Early birds take more risks and get more rewards. Otherwise the network would have problems getting enough miners involved in the first place.
Asides that, there is a concept of inflation and deflation - we have already experienced the first one, so we might as well try the latter one.
Technically, constant rewards, halving rewards… doubling rewards, or any other fluctuation that depends only on the current blockchain state, all are feasible. (Halving rewards guaranties a bound on the total amount of bitcoins in circulation, which helps in avoiding overflows, but this is an extremely minor point that could be overcome very easily.)
Rewards have two unrelated repercussions:
They incentivize mining. This basically means buying hashpower for the network. More rewards means a more secure network, but also more waste in electricity.
They continuously inject new currency shares. This means that, in the long term, all bitcoin holders (not only transactors) pay a fee for the sustained service offered by the network. (In the same way governments, or nowadays banks, collect an inflation “fee” with traditional currencies : with inflation government or bank debts are being forgotten over time.)
First of all, it's unclear as to what amount of hashpower guaraties securrity without being wasted. To make things more difficult, the amount of hashpower that corresponds to a given reward depends on the trade value of the currency. So, Satoshi took a wild guess and he decided halving rewards (every four years) would somehow be satisfactory for the early days of bitcoins. There is no reason for this to remain true in the future. If it was not satisfactory any more, be assured that either Bitcoin workings will change or that another currency with adjusted parameters will quickly be born.
Now, you may realize that Bitcoin only provides one degree of liberty. Rewards must be adjusted to network-maintenance costs, so they cannot be used to influence inflation without being stupidly wasted. This is because Bitcoin does not come with a global community-shared fund (like government or bank reserves) that could receive newly minted money or be cleared from “old debts”. In fact, no ideological choice had to be made, except for the choice not to add such a thing in Bitcoin.
In some ways, halving reward prevents people with power and money to manipulate the bitcoin network. Supposed you have lots of dollars. You decided the bitcoin network is now valuable for your purpose. You can put your money into bitcoin. If there's no halving, price is likely to be stable. Your share of coins will increase over time. If you have the power to print money, you can inflate your currency to do this.
At some point, you have enough coins to control price movement. With halving, you get diminishing return. When you inflate your money to control the network, you get less coins.
I think if it inflates at exponential rate, inflation can also deflate manipulators. But it has to be exponential inflation, not constant inflation.