I think that we need rewards for comleting tiers. It may be treasure boxes, prize boxes or something like mission rewards.

Treasure boxes variant

For completing 1 Tier you get access to room with 20 boxes(depth 8, 13, 29)
For comleting 2 Tiers you get access to the first room with 20 boxes and second with 30 boxes(depth 13,29) BUT, if you visited 20 boxes room before, you will not get access to it, only 30 boxes.
For comlpeting all 3 Tiers you get access to all 3 rooms: 20, 30 and 50 boxes.(depth 29) The same, 20 and 30 boxes rooms are available only at once.
Of course all rooms requires barriers.
Maximum reward is 100 boxes for a full run.
P.S. There are 31 boxes on arena.

Prize boxes variant

For completing One Tier you get bronze Tier1 (2 or 3) prize box
For completing Two Tiers you get silver Tier 2(3) prize box
For comlpeting all Three Tiers you get golden Tier 3 prize box
Maximum reward is 3 boxes(bronze T1, silver T2, golden T3) for a full run. Prize boxes are not available on normal difficulty.

Arcade rewards variant

You know about mission rewards. What if we will get the same for arcade.
There should be 2 values for that:
1 - Number passed prize wheels( not depths, cause terminals mean nothing);
2 - Last depth.
1 value defines size of the prize, 2 - quality of the prize.