The

reward method triangle is a visual way to represent the different tradeoffs when selecting a reward system for a mining pool.

The 3 primary factors which determine the attractiveness of a pool (at least as far as the reward method is concerned), are:

- Expected payout

- Variance of payout

- Maturity time

The expected payout is affected by several external factors which are unrelated to the reward method; however, it is influenced by the risk posed on the pool operator by the reward method, as high risk will cause him to charge a higher fee to compensate. Therefore, the above factors can be translated to the following attributes which the reward system seeks to minimize:

- Variance

- Maturity time

- Operator risk

It is impossible to eliminate all 3 simultaneously; however, by choosing a reward method and its parameters, it is possible to achieve any desired tradeoff between the 3. (In this analysis we disregard some sources of additional variance and maturity time, such as variable block rewards and the 120-block maturation time).

The possible choices can be visualized with the reward method triangle:

Every point in the triangle represents a possible tradeoff between the 3 attributes.

The red vertex represents high variance, while the opposing red edge represents no variance; the more distant a point is from the red edge, the more variance there is in a method corresponding to the point.

Similarly, the green vertex represents high maturity time, with the green edge representing no maturity time; and the blue vertex represents high operator risk, with the blue edge representing no operator risk.

The shaded area corresponds to all possibilities that can be obtained with different parameters for the double geometric method. The methods represented by the edges and vertices are also special cases of DGM. This isn't a special property of DGM per se - other methods can also encompass the entire triangle if they are sufficiently parameterized. You can reduce any attribute as much as you like by moving close to the corresponding edge, but at the cost of increasing the other two; and you can even eliminate any two attributes by approaching the vertex of the third, at the cost of pushing that third attribute to the highest possible value (the blue vertex is a special case which I discuss in detail below).

The red vertex corresponds to solo mining, which has no maturity time or risk, but has the highest possible variance. It is also a special case of DGM with c=0, r=0, of the geometric method with c=0, and of PPLNS with N=1 share.

The blue edge represents PPLNS in all its variants (unit-, exponential-, shift-, pay-once-, but not the naive versions which are not hopping-proof), which has no operator risk. By choosing a value for N (or the equivalent parameter for the specific variant), we select a point along the edge; a low value is closer to the red vertex with high variance and low maturity time, while a high value is closer to the green vertex with low variance and high maturity time. Exponential-PPLNS is a special case of DGM with c=0, o=1 and varying values of r.

As N goes to infinity in PPLNS, we approach the green vertex, which has no variance or risk, but infinite maturity time. This is DGM with o=1, r=1.

The red edge represents DGM with c=1, r=1 and varying values of o. This has no variance, and the tradeoff between maturity time and risk is controlled by o - the lower o is, the greater the risk and the lesser the maturity time. However, even if we set o to 0 and follow the method verbatim, there will still be maturity time. The payouts in this case will be exactly as in PPS, but the method description specifies that they can only be given when the next block is found. This is why the blue vertex, corresponding to PPS (with the highest risk, but no variance or maturity time) stands on its own - we need to explicitly specify that we will give rewards according to PPS rather than following the generic method. This is also why there is a white area which is probably impossible to achieve with any reward method (unless there is a two-way payment channel between the operator and miners).

The black edge which is adjacent to the green edge represents the geometric method. Its maturity time is not 0, which is why it does not coincide with the green edge; but it has the lowest maturity time for any given generic tradeoff between risk and variance. The higher the parameter c is, the lower the variance and the higher the risk. As c approaches 1, we approach the intersection between the black and red edges, which is equivalent to PPS for all purposes except maturity time.