I am wondering if anyone can give me a GTO solution (if one exists) on this game.

The game has twenty players and consists of multiple rounds. In each round there will be ten heads up matches. In each match one player is designated the dealer and the other player is designated the runner. The two players will be given a total of 100 chips which will be divided between the two players however the DEALER chooses.

The catch is, the runner must accept the dealers offer otherwise neither player gets any chips.

We will play each opponent twice (once as the dealer and once as the runner).

What is our best strategy? How will this change if our opponent is always unknown to us?

My first thought is that the utilities aren't defined very well.

What happens at the end of this game? Do all players exchange all of their chips for cash? Is there a bonus for having the most chips? Is there only one "winner", i.e. the chip leader after 20 rounds? There are many other such questions left unanswered that may affect a Nash equilibrium for this game.

I'm not very good at solving games, so maybe I'm missing some subtlety in the rules or payoffs, but if I've understood OP correctly...
Every dealer says "I am taking all 100 chips".
Every runner says "I have to accept that".
It doesn't seem like much of a game if the runner has no strategic options. :/

Actually if the dealer said he was keeping all 100 chips, the runner could refuse because he gets 0 either way.

But if the dealer said "I want 99 chips and you get 1", then it looks as though the runner has to take that. What I was wondering in my first post is if there's some externality that the OP didn't mention that would make that strategy not a Nash equilibrium (i.e. giving the dealers some incentive to offer less as part of a metagame strategy).

yeah, Vernon seems right, if the payouts are simply the chips each player receives in her 18 matches, then 99,1 seems like the solution; at 100 the dealer has to fear getting punished by the runner (which she can do because he gets zero either way), at 98 the dealer is giving up 1 chip because the runner has to accept either way since 1>0.

the game would be more interesting if the repeated interaction entailed some effective punishment mechanism in the 2nd match of two respective players; as it stands it doesn't really matter whether you play 9 opponents twice or 18 different opponents.

edit: ^^ i thought there are 10 players total, but it doesn't matter anyways

Actually there is one way that the multiplayer nature of this game could make it interesting--but if the chips are just straight cash, and the place order of chip counts doesn't matter, then it's not really relevant. The idea is that (if no one else is colluding) two players could collude by refusing all chip splits with all the other players, thereby guaranteeing themselves the 1 and 2 spot in the chip count.

However, if the chips are straight cash, then the colluders are actually just costing themselves money by doing this, since each round is positive-sum for all players.

Yeah I think CallMeVernon is right 99,1 seems to be the solution. I've seen some Nash videos on YouTube way back in 2012 when I was looking up Nash stuff. There was a video that said something similar to 99,1 being the solution. The only reason in real life why someone wouldn't want to accept the 1 chip is out of spite, but it is still better than 0 if we look at the game from a theoretical perspective.

If each chip is worth 1 million dollars in cash I'm sure no one would be doing any spite plays.

To make the game more interesting I would suggest there be two dealers and 1 runner. That way there are two deals made where each dealer is aware of who the other dealer is, but unaware of what deal the other dealer is offering. The runner will then want to select the best offer. If both offers are exactly the same (or too close in value) then the runner can exclaim "monkey money thief". Both dealers then collect nothing and the runner collects some type of bonus prize like 25 chips or whatever. This little caveat may scare some dealers from low balling too much. If both deals are nicer than 25 chips the runner won't have much incentive to say "monkey money thief" since accepting a deal may be better.

On second thought "monkey money thief" might not be necessary at all since they are both competing for runner to make a deal. Whatever, there is a lot you can do with this game. How it is right now is too easy for the dealer to play correctly.

A far better version of this game is if the dealer writes down (in secret) how much he wants to give the runner, the runner writes down how much he wants, and if the numbers don't match exactly they both get 0, but if they do then the runner gets that amount and the dealer keeps the rest. It turns out that in this format, writing down any number from 1 to 99 could be considered "GTO".