We begin with a deck of cards in which 50% are Aces (you can use Red cards for Aces) and 50% are Kings (you can use Black cards for Kings). There are two players and one dealer. The play begins by each player putting in the ante (1 chip). Each player is dealt one card face down. WITHOUT LOOKING AT THEIR CARD, the players decide to Bet (say, 1 chip) or Fold. Players secretly show the dealer their choice of Bet or Fold. If one player Bets and the other Folds, then the player who Bet wins. If both Bet or both Fold, then Ace beats King (or Red beats Black); winner takes the pot (all the chips from the ante and any bets). If there is a tie, they split the pot.
Play the game several times with two other people (so you have two players and a dealer). Keep track of the strategy choices of the players and the resulting payoffs.
Before moving on, you should attempt to determine the payoff matrix. The remainder of this section will be more meaningful if you have given some thought to what the payoff matrix should be. It is OK to be wrong at this point, it is not OK to not try.
If Player 1 Bets and Player 2 Folds, does it matter which cards were dealt? How much does Player 1 win? How much does Player 2 lose? What is the payoff vector for [Bet, Fold]? (Keep in mind your answer to Activity 2.4.3.)
To determine the payoff vector for [Bet, Bet] and [Fold, Fold] we will need to consider which cards were dealt. We can use some probability to determine the remaining payoff vectors.
There are four possible outcomes of the deal (what cards could have been dealt to each player?). List them. What is the probability that each occurs? Remember, the probability of an event is a number between 0 and 1.
Activity2.4.11.Payoff for each deal with [Bet, Bet].
Consider the pair of strategies [Bet, Bet]. For each possible deal, determine the payoff vector. For example, if the players are each dealt an Ace (Red), how much does each player win? Again, keep in mind your answer to Activity 2.4.3.
In order to calculate the payoff for [Bet, Bet], we need to take a weighted average of the possible payoff vectors in Activity 2.4.11. In particular, we will “weight” a payoff by the probability that it occurs. Recall that this is the expected value, Definition 2.3.4. We will calculate the expected value separately for each player.
Now that you have done all the hard work of finding the payoff matrix for One-Card Stud Poker, we can analyze our two-player zero-sum game using the techniques we learned in the previous sections. It is also important to see how the mathematical solution compares to our conjectured solution from Activity 2.4.2.
Compare the strategy you found in Activity 2.4.17 to your suggested strategy in Activity 2.4.2. In particular, discuss how knowing the payoff matrix might have changed your strategy. Also compare the payoff that results from the strategy in Activity 2.4.17 to the payoff that results from your original strategy in Activity 2.4.2.
Since One-Card Stud Poker has an element of chance, we should see what happens if we play the game several times using the strategy from Activity 2.4.17.
Play the game ten times using the best strategy. How much has each player won or lost after ten hands of One-Card Stud Poker? Compare your answer to your prediction in Activity 2.4.19. Does the actual payoff differ from the theoretical payoff? If so, why do you think this might be?
In One-Card Stud Poker we anted one chip and bet one chip. Now, suppose we let players ante a different amount and bet a different amount (although players will still ante and bet the same amount as each other). Suppose a player antes \(a\) and bets \(b\text{.}\) How might this change the game?
To make this a little easier, assume the deck has infinitely many cards, so that the probability of a player being dealt an Ace doesn’t change if the other player was dealt an Ace. In other words, each player has a probability of \(.25\) of being dealt an Ace. Now the probability of the deal Ace, Ace is \(.25\times .25\text{.}\)
Now that we have analyzed several zero-sum games, we can see how important it is to find any equilibrium pairs. In the next section we explore equilibrium strategies in more detail.
In a game of chance the actual outcome is often different that the expected outcome. As the game is played more, the actual payoff approaches the expected payoff.
False.
In a game of chance the actual outcome is often different that the expected outcome. As the game is played more, the actual payoff approaches the expected payoff.
