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The dollar auction is a type of auction in which players bid money for a dollar. Whoever bids the most pays what they bid and gets the dollar. Whoever bids the second most pays what they bid and get nothing. If the players follow a naive strategy, everyone will try to bid 1 cent more than the current bid until it hits 99 cents, after which the top two players will keep overbidding each other ad infinitum.

This however would obviously not be a Nash equilibrium, since the top two players would switch to not bidding at all.

However, everyone not bidding at all is also not a Nash equilibrium, since then any individual player would benefit from deciding to bid, since then they get a dollar for a penny. My question is, what are the (sub-game perfect) Nash equilibrium then?

Here are some simplifications:

  1. So that one of the players does not immediately bid 99 cents, the players can only bid 1 cent at a time.
  2. There will only be two players. They will alternate turns. On each turn, they can either bid one more cent than the other player (or one cent on the first turn), or not bid.
  3. The first player to not bid pays their highest bid (or 0 if they did not bid) and gets nothing. The other player pays their highest bid and gets a dollar (100 cents). (Note: Yes, this means if the first player passes on the first turn, the second player gets a free dollar).
  4. As it stands, this game is not well defined since it could go on forever. To prevent this (and also make the number of choices finite), there will be maximum bid. After the bid reaches a million dollar (100,000,000 cents), a fair coin is flipped. If it heads, the next player must not bid. If tails, the next player may bid, but if they do, then the other player must not bid. (The coin ensures symmetry.)
  5. The players' payoffs are just their gains minus their loses (i.e. their utility is a linear function of their wealth).

Since this is now a finite extensive game, we know a sub-game perfect Nash equilibrium will exist. What are these equilibrium?

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