Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

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: The hybrid equilibrium is when Player 1 stays with probability y and folds with probability (1 - y), while Player 2 folds with probability 11/23 and calls with probability 12/23. The players' expected payoffs are as follows: if Player 1 stays, their expected payoff is approximately 8.27, and if they fold, their expected payoff is -2.50. Player 2's expected payoffs are -8.27 if Player 1 stays and 2.50 if Player 1 folds.

Step by step solution

01

1. Identify the Actions

In the game, player 1 can either stay or fold. If player 1 stays, player 2 can then either fold or call. Thus, there are four possible actions: player 1 staying and player 2 folding, player 1 staying and player 2 calling, player 1 folding, and player 2 folding.
02

2. Determine the Outcomes

After identifying the possible actions, we can determine the outcomes of the game: - If player 1 folds, player 1 pays player 2 $2.50. - If player 1 stays, and - if player 2 folds, player 2 pays player 1 $1.50. - if player 2 calls and - if the card is low (2 to 8), player 2 pays player 1 $100. - if the card is high (9 to Ace), player 1 pays player 2 $100.
03

3. Draw the Extensive Form

Now, we can draw the extensive form, showing the different actions and outcomes for each player: 1. Player 1 (stay/fold) - Stay: - Player 2 (fold/call) - Fold: pay $1.50 to Player 1 - Call: - Low card: pay $100 to Player 1 - High card: pay $100 to Player 2 - Fold: pay $2.50 to Player 2 Note that the extensive form represents the possible decisions made by the players, and the outcomes following each possible set of decisions. #b. Solve for the hybrid equilibrium.#
04

1. Determine the Strategies

To find the hybrid equilibrium, we need to determine the mixed Nash equilibrium for the second player and compare it with the pure strategies of player 1 (staying or folding). First, let x be the probability of player 2 folding, and 1 - x be the probability of player 2 calling. We also let y be the probability of player 1 staying and 1 - y be the probability of player 1 folding.
05

2. Calculate Players' Payoffs

Let's calculate the players' payoffs when Player 1 stays. - Expected Payoff for Player 1: \((7/13) \cdot x \cdot 100 + (6/13) \cdot (1 - x) \cdot (-100) - (1 - x) \cdot 2.50\) - Expected Payoff for Player 2: \((7/13) \cdot x \cdot (-100) + (6/13) \cdot (1 - x) \cdot 100 - x \cdot 1.50\) Now we need to find the best response for player 1 given player 2's strategy. It can be done by comparing player 1's payoffs for staying and folding: \([100x - 6(1 - x)] \ge 2.50\)
06

3. Find the Hybrid Equilibrium

Solving for x, we get: \(x = \frac{11}{23}\) So, the hybrid equilibrium would be: - Player 1: Stay with probability y, Fold with probability (1 - y) - Player 2: Fold with probability \(\frac{11}{23}\), Call with probability \(\frac{12}{23}\) Note that in the hybrid equilibrium, player 2 adopts a mixed strategy, while player 1 can choose either a pure strategy (stay or fold) or a mixed strategy depending on their preferences. #c. Compute the players' expected payoffs.# Using the hybrid equilibrium strategy for player 2, we can compute the expected payoffs for both players. First, we calculate the expected payoffs when player 1 folds: - Player 1's expected payoff: \(-2.50\) - Player 2's expected payoff: \(2.50\) Now, we calculate the expected payoffs when player 1 stays: - Player 1's expected payoff when staying: \((7/13) \cdot (11/23) \cdot 100 + (6/13) \cdot (12/23) \cdot (-100) - (12/23) \cdot 2.50 \approx 8.27\) - Player 2's expected payoff when staying: \((7/13) \cdot (11/23) \cdot (-100) + (6/13) \cdot (12/23) \cdot 100 - (11/23) \cdot 1.50 \approx -8.27\) Hence, if player 1 decides to stay, their expected payoff is about \(8.27, and if they decide to fold, their expected payoff is -\)2.50, while player 2's expected payoffs are -\(8.27 and \)2.50, respectively.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Understanding Extensive Form Games
Extensive form games are a type of game representation that offers a way to visualize the sequential nature of games. This graphical depiction includes decision nodes, branches representing the choices available to the players, and the consequences of these choices.
In our example of Blind Texan Poker, we begin with Player 1's decision to stay or fold. If Player 1 stays, the decision moves to Player 2, with further branches based on calling or folding, and then down to the outcomes dependent on the card revealed. Such detailed representation of the game allows for a deep analysis of strategies and payoffs, showcasing the decisions at each point.

Key Points in an Extensive Form Game:

  • Decision points indicate where a player must make a choice.
  • Terminal nodes indicate final outcomes with associated payoffs.
  • Information sets reflect what a player knows at the point of making a decision.
Understanding the structure of the extensive form game is crucial for strategizing and foreseeing the potential outcomes of each action taken by the players.
Mixed Nash Equilibrium in Game Theory
A Mixed Nash Equilibrium is a strategic concept wherein players randomize their strategies to keep their opponents indifferent between their possible actions. It's a situation where no player can benefit by unilaterally changing their strategy when the strategies of others are given.
In the Blind Texan Poker example, when we talk about solving for the hybrid equilibrium, we're essentially looking for a mixed Nash equilibrium for Player 2. This equilibrium occurs when Player 2 folds with some probability and calls with the complement of that probability. Player 1's best response to this mixed strategy is found by calculating and comparing expected payoffs from staying or folding and choosing the action that maximizes their payoff.

Characteristics of Mixed Nash Equilibrium:

  • Players use probabilities to determine their actions.
  • Each player's strategy is a best response to the strategies of others.
  • Mixed strategies are used when there are no pure strategy Nash equilibria.
This equilibrium concept enables players to plan their actions under uncertainty, considering that their opponents also strategize intelligently based on available information.
Calculating Expected Payoffs
Expected payoffs are a fundamental aspect of game theory, allowing players to evaluate the average outcome of their decisions in probabilistic terms. An expected payoff is calculated by multiplying the payoff of each possible outcome by the probability of that outcome, then summing these products.
In the context of Blind Texan Poker, we determine Player 1 and Player 2's expected payoffs based on the hybrid equilibrium strategies. For example, if Player 1 stays, we calculate the sum of the product of the probability of Player 2 folding (and Player 1 winning) and the probability of Player 2 calling with either a high card or a low card and the accompanying win or loss. These calculations provide insights on whether staying or folding has a higher expected value for the players.

Factors Influencing Expected Payoffs:

  • Possible outcomes of a player's actions.
  • The probabilities of these outcomes occurring.
  • The payoff associated with each outcome.
By understanding how to compute expected payoffs, players can make informed decisions that maximize their average winnings over time, factoring in the strategies that their opponents may use.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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 both his own income \(Y_{1}(r)\) and the income of his parent \(Y_{2}(r),\) where \(Y_{1}^{\prime}(r)>0\) and \(Y_{2}^{\prime}(r)<0 .\) Later, the parent moves, leaving a monetary bequest \(L\) to the child. The child cares only for his own utility, \(U_{1}\left(Y_{1}+L\right),\) but the parent maximizes \(U_{2}\left(Y_{2}-L\right)+\alpha U_{1},\) 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 \(Y_{1}+Y_{2}\) 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 (NE) 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 \(102(1987): 179-221\)

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.

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?
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free