Where k is the number of frames in a timestep when getting successors.  So instead of MarioState ms = next.next(a, next.ws); that is put into a loop where ms is the result of applying next.next(a) k times.

k = 1
Scoring controller ch.idsia.ai.agents.ai.TimingAgent@70329f3d with starting seed 0
Difficulty 0 score 4321.6000 (avg time 5.3836)
Difficulty 3 score 4296.0000 (avg time 6.5108)
Difficulty 5 score 4371.2000 (avg time 6.8178)
Difficulty 10 score 4320.0000 (avg time 11.7151)
Competition score: 17308.8

k = 2
Scoring controller ch.idsia.ai.agents.ai.TimingAgent@59bca5f1 with starting seed 0
Difficulty 0 score 4321.6000 (avg time 2.3875)
Difficulty 3 score 4296.0000 (avg time 3.4120)
Difficulty 5 score 4371.2000 (avg time 3.9065)
Difficulty 10 score 4023.5445 (avg time 5.1856)
Competition score: 17012.344506835936

k = 3
Scoring controller ch.idsia.ai.agents.ai.TimingAgent@593d93f4 with starting seed 0
Difficulty 0 score 4321.6000 (avg time 4.1105)
Difficulty 3 score 3465.3356 (avg time 5.5774)
Difficulty 5 score 4371.2000 (avg time 5.2654)
Difficulty 10 score 4320.0000 (avg time 6.2693)
Competition score: 16478.135595703126

k = 4
Scoring controller ch.idsia.ai.agents.ai.TimingAgent@59bca5f1 with starting seed 0
Difficulty 0 score 3927.7374 (avg time 5.2220)
Difficulty 3 score 4166.9405 (avg time 7.0790)
Difficulty 5 score 4371.2000 (avg time 6.9685)
Difficulty 10 score 4320.0000 (avg time 7.8314)
Competition score: 16785.87783203125

k = 5
Scoring controller ch.idsia.ai.agents.ai.TimingAgent@59bca5f1 with starting seed 0
Difficulty 0 score 596.8327 (avg time 9.0660)
Difficulty 3 score 1260.5181 (avg time 9.9126)
Difficulty 5 score 913.7884 (avg time 8.1610)
Difficulty 10 score 2120.9174 (avg time 7.1636)
Competition score: 4892.056545257568