Let’s play another simple card game: after a $1 ante, you (the dealer) shuffle a standard 52-card poker deck, and deal one card face down to each of us. After looking at our respective cards, we each have the option either to “stay” and keep our card, or to “switch,” discarding and drawing a new card from the top of the deck. The player with the highest-ranked card (ace low through king high) wins the pot, where a tie splits the pot. What is the optimal strategy and expected value of playing this game?
(Note that although all of the games discussed here are playable– and are even more interesting and complex– with three or more players, I am focusing on just two players here.)
There are two interesting variations on these basic rules. First, suppose that we declare our choice of whether to stay or switch simultaneously, as in the game Guts: each player takes a chip or bottle cap under the table, either places it in a closed fist (to stay) or not (to switch), then holds the closed fist over the table. All players simultaneously open their hands palm-down to indicate their choice.
This is the simpler version of the game. With two players, it can be shown that the optimal strategy for both players is to stay if they are dealt a nine or higher. That the game is fair (i.e., the expected value is zero) is clear from symmetry; no one player is “preferred” or distinguished in any way by the rules of the game.
One-card draw poker
Now suppose instead that we play more like a typical draw poker game: after the initial deal, each player in turn decides whether to discard and draw a new card. Remember that you are the dealer, so you get to make your decision after I have made mine. Can you exploit this advantage?
This is the version of the game that motivated this post. I came up with this while trying to find examples of games for a student, games that were “small” enough to be tractable to analyze and experiment with via simulation, but still with interesting strategic complexity. It’s worth noting that the name “One-Card Poker” is also used to refer to a similar “stud” form of the game, where the card play is simpler (no discarding), but the betting rounds are as in normal poker, resulting in a game that is much more complex to analyze.
(One final note: use of an actual 52-card deck complicates the analysis slightly, in a mostly non-interesting way. I find it convenient to present the game using an “infinite deck,” where the probability distribution of card ranks remaining in the deck does not change as cards are dealt.)