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Chapter 8: Problem 8

In Blind Texan Poker, player 2 draws a card from a standard deck and places it against her forehead without looking at it but so player 1 can see it. Player 1 moves first, deciding whether to stay or fold. If player 1 folds, he must pay player 2 \$50. If player 1 stays, the action goes to player 2 . Player 2 can fold or call. If player 2 folds, she must pay player \(1 \$ 50\). If player 2 calls, the card is examined. If it is a low card \((2-8)\), player 2 pays player \(1 \$ 100\). If it is a high card \((9,10, \text { jack, queen, king, or ace), player } 1\) pays player \(2 \$ 100\) a. Draw the extensive form for the game. b. Solve for the hybrid equilibrium. c. Compute the players' expected payoffs.

Short Answer

Expert verified
Answer: In the hybrid equilibrium, the expected payoffs for player 1 and player 2 are \$0 and \$4.62, respectively.

Step by step solution

01

In Blind Texan Poker, player 1 can see the card drawn by player 2 but player 2 cannot. Player 1 decides whether to stay or fold. If player 1 folds, he pays player 2 \$50. If player 1 stays, player 2 decides whether to fold or call. If player 2 folds, she pays player 1 \$50. If player 2 calls, the card is examined, and the payoff depends on the card being high or low, as described in the problem. Analyze this game step by step, deducing the possible strategies and outcomes. #Step 2: Draw the Extensive Form#

Draw the extensive form of the game by representing each player's possible decisions in the form of a decision tree. The nodes represent the decision points for each player, with the possible moves branching off from these points. The terminal nodes represent the final outcomes of the game, with their associated payoffs. #Step 3: Identify the Hybrid Equilibrium#
02

A hybrid equilibrium is a mixture of Nash equilibria and Bayesian equilibria. In this case, it means determining the probabilities of each player choosing the action "call" in a way that these probabilities make them indifferent between calling and folding, i.e., the expected payoffs are equal for these actions. For player 1, let p be the probability of calling with a high card: p * (-\$100) + (1-p) * (\$100) = 0 Solve for p to obtain the probability for player 1: p = 1/2 For player 2, let q be the probability of calling when player 1 stayed: \frac{7}{13} * q*(-\$100) + \frac{6}{13} * q*(\$100) = -\$50 Solve for q to obtain the probability for player 2: q = \frac{10}{13} #Step 4: Compute the Players' Expected Payoffs#

To compute the expected payoffs for each player, consider the possible actions and the probabilities of all outcomes based on the hybrid equilibrium: Expected payoff for player 1: - If player 1 folds, he loses \$50. - If player 1 stays with a high card (probability = 6/13), his expected payoff is 0.5 * (\$100) - 0.5 * (\$100) = 0. - If player 1 stays with a low card (probability = 7/13), his expected payoff is \frac{10}{13} * (\$50) + \frac{3}{13} * (-\$50) = \$25. Player 1's overall expected payoff is 0 (since he can choose to fold when he sees a high card). Expected payoff for player 2: - If player 2 calls (probability = 10/13), her expected payoff is \frac{7}{13} * (-\$100) + \frac{6}{13} * (\$100) = -\$15.38. - If player 2 folds (probability = 3/13), her expected payoff is \$50. Player 2's overall expected payoff is the weighted average of calling and folding payoffs: \(\frac{10}{13} * (-\$15.38) + \frac{3}{13} * (\$50) = \$4.62\). In summary, the expected payoffs for player 1 and player 2 in Blind Texan Poker's hybrid equilibrium are \$0 and \$4.62, respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Extensive Form Game
The concept of an extensive form game is central to understanding strategic interactions in scenarios such as the Blind Texan Poker game. It is represented through a decision tree which illustrates all possible moves, decisions, and outcomes within a game.

An extensive form diagram visualizes the sequence of play, including the choices available to each player at every decision point. The nodes represent the decision points, the branches are the actions taken by the players, and the terminal nodes indicate the outcomes with associated payoffs.

In this poker game, the extensive form highlights the two-stage decision process: first, player 1's choice to stay or fold, and then, based on player 1's action, player 2's decision to fold or call. Clear visualization in such a decision tree can be particularly helpful, as it allows the players to anticipate possible outcomes and strategically decide their moves.
Hybrid Equilibrium
Understanding the hybrid equilibrium concept is key to analyzing mixed-strategy games like Blind Texan Poker. A hybrid equilibrium is where players randomize their strategies, blending components of Nash and Bayesian equilibria.

In this context, each player calculates the probability of making certain moves that leave them indifferent between their available options. These probabilities ensure that players maximize their utility considering the uncertainty of their opponents' strategies.

For example, player 1 must find a balance in the probability of calling with a high card, and player 2 must determine the likelihood of calling after player 1 stays. The hybrid equilibrium reflects the reality of poker, where bluffing and unpredictability are crucial aspects of the game. Here, the equilibrium provides a sophisticated model of player behavior, capturing the essence of playing strategies where full information is not available.
Expected Payoffs
Calculating the expected payoffs in games like Blind Texan Poker involves examining the potential outcomes and their associated probabilities. It allows players to quantify the expected value of their decisions, providing a mathematical foundation for their strategic choices.

In the poker example, we consider the hybrid equilibrium probabilities and compute the expected value for each player's actions. This calculation involves summing the products of the payoffs and their probabilities for every possible result of the game. Since utilitarian outcomes and strategies are the core of microeconomic game theory, the computation of expected payoffs lies at the heart of players' decision-making process.

By mastering expected payoffs, players can make more informed choices, avoiding suboptimal moves and enhancing their chances of achieving the best possible results given the inherent risks and uncertainties of the game.

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Most popular questions from this chapter

Consider a simple version of the Ultimatum Game discussed in the text. The first mover proposes a division of \(\$ 1 .\) Let \(r\) be the share received by the other player in this proposal (so the first mover keeps \(1-r\) ), where \(0 \leq r \leq 1 / 2\). Then the other player moves, responding by accepting or rejecting the proposal. If the responder accepts the proposal, the players are paid their shares; if the responder rejects it, both players receive nothing. Assume that if the responder is indifferent between accepting or rejecting a proposal, he or she accepts it. a. Suppose that players only care about monetary payoffs. Verify that the outcome mentioned in the text in fact occurs in the unique subgame-perfect equilibrium of the Ultimatum Game. b. Compare the outcome in the Ultimatum Game with the outcome in the Dictator Game (also discussed in the text), in which the proposer's surplus division is implemented regardless of whether the second mover accepts or rejects (so it is not much of a strategic game!). c. Now suppose that players care about fairness as well as money. Following the article by Fehr and Schmidt cited in the text, suppose these preferences are represented by the utility function \\[ U_{1}\left(x_{1}, x_{2}\right)=x_{1}-a\left|x_{1}-x_{2}\right| \\] where \(x_{1}\) is player 1 's payoff and \(x_{2}\) is player 2 's (a symmetric function holds for player 2 ). The first term reflects the usual desire for more money. The second term reflects the desire for fairness, that the players' payoffs not be too unequal. The parameter \(a\) measures how intense the preference for fairness is relative to the desire for more money. Assume \(a<1 / 2\) 1\. Solve for the responder's equilibrium strategy in the Ultimatum Game. 2\. Taking into account how the second mover will respond, solve for the proposer's equilibrium strategy \(r^{*}\) in the Ultimatum Game. (Hint: \(r^{*}\) will be a corner solution, which depends on the value of \(a\).) 3\. Continuing with the fairness preferences, compare the outcome in the Ultimatum Game with that in the Dictator Game. Find cases that match the experimental results described in the text, in particular in which the split of the pot of money is more even in the Ultimatum Game than in the Dictator Game. Is there a limit to how even the split can be in the Ultimatum Game?

Two neighboring homeowners, \(i=1,2,\) simultaneously choose how many hours \(l_{i}\) to spend maintaining a beautiful lawn. The average benefit per hour is \\[ 10-l_{i}+\frac{l_{j}}{2} \\] and the (opportunity) cost per hour for each is \(4 .\) Homeowner \(i\) 's average benefit is increasing in the hours neighbor \(j\) spends on his own lawn because the appearance of one's property depends in part on the beauty of the surrounding neighborhood. a. Compute the Nash equilibrium. b. Graph the best-response functions and indicate the Nash equilibrium on the graph. c. On the graph, show how the equilibrium would change if the intercept of one of the neighbor's average benefit functions fell from 10 to some smaller number.

Return to the game with two neighbors in Problem \(8.5 .\) Continue to suppose that player \(i\) 's average benefit per hour of work on landscaping is \\[ 10-l_{i}+\frac{l_{j}}{2} \\] Continue to suppose that player 2 's opportunity cost of an hour of landscaping work is \(4 .\) Suppose that player 1 's opportunity cost is either 3 or 5 with equal probability and that this cost is player 1 's private information. a. Solve for the Bayesian-Nash equilibrium. b. Indicate the Bayesian-Nash equilibrium on a best-response function diagram. c. Which type of player 1 would like to send a truthful signal to player 2 if it could? Which type would like to hide his or her private information?

In \(A\) Treatise on the Family (Cambridge, MA: Harvard University Press, 1981 ), Nobel laureate Gary Becker proposes his famous Rotten Kid Theorem as a sequential game between the potentially rotten child (player 1 ) and the child's parent (player 2 ). The child moves first, choosing an action \(r\) that affects his own income \(\Upsilon_{1}(r)\left[Y_{1}^{\prime}(r)>0\right]\) and the income of the parent \(\Upsilon_{2}(r)\left[\Upsilon_{2}^{\prime}(r)<0\right] .\) Later, the parent moves, leaving a monetary bequest \(L\) to the child. The child cares only for his own utility, \(U_{1}\left(\Upsilon_{1}+L\right),\) but the parent maximizes \(U_{2}\left(\Upsilon_{2}-L\right)+\alpha U_{1},\) where \(\alpha>0\) reflects the parent's altruism toward the child. Prove that, in a subgame-perfect equilibrium, the child will opt for the value of \(r\) that maximizes \(\Upsilon_{1}+\Upsilon_{2}\) even though he has no altruistic intentions. Hint: Apply backward induction to the parent's problem first, which will give a first-order condition that implicitly determines \(L^{*} ;\) although an explicit solution for \(L^{*}\) cannot be found, the derivative of \(L^{*}\) with respect to \(r-\) required in the child's first-stage optimization problem-can be found using the implicit function rule.

Recall the job-market signaling game in Example 8.9 a. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education \((N E)\) and where the firm offers an uneducated worker a job. Be sure to specify beliefs as well as strategies. b. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education \((N E)\) and where the firm does not offer an uneducated worker a job. What is the lowest posterior belief that the worker is low-skilled conditional on obtaining an education consistent with this pooling equilibrium? Why is it more natural to think that a low-skilled worker would never deviate to \(E\) and thus an educated worker must be high-skilled? Cho and Kreps's intuitive criterion is one of a series of complicated refinements of perfect Bayesian equilibrium that rule out equilibria based on unreasonable posterior beliefs as identified in this part; see I. K. Cho and D. M. Kreps, "Signalling Games and Stable Equilibria," Quarterly Journal of Economics \(102(1987): 179-221\)
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