In particular, suppose if you mined normally, you’d extract fees a, b from the blocks, but if you add the tx, you instead extract fees a_1, b_1 where a_1 < a and a_1 + b_1 > a+b. In that case, call hashrate mining a_1 “strategic” and say their hashrate is is 0 < h < 1, then the payoff matrix is:
| probability | who mines block a, b | block a value | block b value | payoff (strategic, normal) |
|---|---|---|---|---|
| h^2 | strategic, strategic | a_1 | b_1 | (a_1+b_1, 0) |
| (1-h)h | normal, strategic | a | b | (b, a) |
| h(1-h) | strategic, normal | a_1 | b_1 | (a_1, b_1) |
| (1-h)^2 | normal, normal | a | b | (0, a+b) |
Expected payoff for strategic miners is h ( a_1 + h\cdot b_1 + (1-h)b )
Expected payoff for normal miners is (1-h) ( a + h\cdot b_1 + (1-h) b )
But once you divide that by their respective hashpower, the proportional payoff for normal miners is higher, as a > a_1.