I'll chime in here with my two sats.

The game theory of on-chain payments is "if you want your payment confirmed in the next block, you have to outbid all of the other transactions in the mempool" but on-chain transactions are not the only way to send bitcoin so the game theory is more about on-chain payments competing with alternative scaling solutions.

IMHO, the most important game theoretical aspect of on-chain payments is that fee spikes incentivize the innovation, investment, and development in alternative scaling methods.

There is no on-chain incentive to lower fees. The only way that the fee rate goes is up. Every sender is bidding up the price with each and every tx they send.

All of the other scaling solutions put downward pressure on the cost of sending payments by taking transactions out of the mempool. Every transaction sent on lightning is a transaction NOT sent on-chain.

There is even game theory within other scaling methods like lightning. For example: lightning routing fees are a bidding war between routing nodes to bid lower than the cost of other routing nodes so fees trend downward over time. I wrote an entire article on the game theory of the lightning network (https://www.whatisbitcoin.com/lightning-network/game-theory-lightning-network)

The higher on-chain fees go, the greater the need/want for investment in other scaling technologies because those new methods are now cost effective whereas they weren't with 1sat/vbyte txs.