Good examples of zero-sum perfect information game?

Question

I am trying to come up with some good examples of zero-sum perfect information games. What I have on the list are:

1. Tic-Tac-Toe
2. Connect $4$
3. Chess

Any thing more?

---

One more question, for Connect4, this game can be solved perfect (1st player is gauranteed a win). If that's the case, can we still come up with a Nash-equilibrium for this game?

Answer 1

You can look at: "Winning ways for your mathematical plays" volume 1, 2, and 3. This is a book from Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy that analyzes many different games. Other games include,

https://en.wikipedia.org/wiki/Hex_(board_game)

https://en.wikipedia.org/wiki/Nim

https://en.wikipedia.org/wiki/Game_of_the_Amazons

https://en.wikipedia.org/wiki/Go_(game)

https://en.wikipedia.org/wiki/Sylver_coinage

Checkers

https://en.wikipedia.org/wiki/Reversi

For board games, getting the Nash Equilibrium (NE) is difficult. The optimal strategy for player $i$ would be a function $f_i$ that would map the board position to an optimal move. Since the number of board positions can be extremely large this function might be very difficult or impossible to obtain. For simple games like tic-tac-toe, you can make a lookup table for each player and each position, this table would be the NE.

Answer 2

The concept of "Nash equilibrium" is not really useful for perfect information games. The whole point of Nash equilibrium is that when you have imperfect information, you need to find suitable probabilities for a mixed strategy. If the game has perfect information, an optimal strategy will be a pure strategy (100% for some move and 0% all other moves).