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Iterated elimination of strictly dominated strategies involves identifying a str...
May 22, 2024
Iterated elimination of strictly dominated strategies involves identifying a strategy that dominates another strategy. The latter task often requires more than a brief inspection of payoffs; one may need to “compute” a mixed strategy that dominates another (pure) strategy, on the basis of relative magnitudes of payoffs. To illustrate this point, consider the following two-player game: 2 LCR U 1,3 4,2 3,5 1 M 2,4 2,0 2,2 D 4,2 1,4 3,0 Consider a mixed strategy αU +(1−α)D. Find the values of α such that αU +(1−α)D strictly dominates M for player 1.
Solution by Steps
step 1
We start by setting up the inequalities for the mixed strategy αU+(1−α)D\alpha U + (1-\alpha)D to strictly dominate MM for player 1. This means that the expected payoff of αU+(1−α)D\alpha U + (1-\alpha)D must be greater than the payoff of MM for each of player 2's strategies (L, C, R)
step 2
For strategy L: \alpha \cdot 1 + (1-\alpha) \cdot 4 > 2 Simplifying, we get: 1\alpha + 4 - 4\alpha > 2 \implies -3\alpha + 4 > 2 \implies -3\alpha > -2 \implies \alpha < \frac{2}{3}
step 3
For strategy C: \alpha \cdot 4 + (1-\alpha) \cdot 1 > 2 Simplifying, we get: 4\alpha + 1 - 1\alpha > 2 \implies 3\alpha + 1 > 2 \implies 3\alpha > 1 \implies \alpha > \frac{1}{3}
step 4
For strategy R: \alpha \cdot 3 + (1-\alpha) \cdot 3 > 2 Simplifying, we get: 3\alpha + 3 - 3\alpha > 2 \implies 3 > 2 This inequality is always true and does not provide any additional constraints on α\alpha
step 5
Combining the results from steps 2 and 3, we get: \frac{1}{3} < \alpha < \frac{2}{3}
Answer
\frac{1}{3} < \alpha < \frac{2}{3}
Key Concept
Mixed Strategy Dominance
Explanation
To find the values of α\alpha such that the mixed strategy αU+(1−α)D\alpha U + (1-\alpha)D strictly dominates MM, we set up and solve inequalities for each of player 2's strategies. The solution is the range of α\alpha that satisfies all the inequalities.
Below we solve, in several steps, for a Nash equilibrium of the following game: 2 LCR T 7,2 2,7 3,6 1 B 2,7 7,2 4,5 Let p ∈ [0, 1] be the probability that player 1 plays T . Let q ∈ [0, 1] be the probabil- ity that player 2 plays L and r ∈ [0, 1] the probability that player 2 plays C . Then a mixed strategy profile is represented by (p, q, r), where q + r ≤ 1. (a) Find player 1’s best response b1(q, r). (b) Find player 2’s best response b2(p). (Hint: For each p ∈ [0,1], check which pure strategies of player 2 give the highest payoff.) (c) Sketch the two best responses b1(·) and b2(·) in the three-dimensional space whose axes are p, q, and r. (d) Using your answer to part (c), find all Nash equilibria of the game.
Solution by Steps
step 1
To find player 1’s best response b1(q,r)b_1(q, r), we need to calculate the expected payoffs for player 1 when playing TT and BB given the probabilities qq and rr for player 2
step 2
The expected payoff for player 1 when playing TT is: E(T)=7q+2r+3(1−q−r) E(T) = 7q + 2r + 3(1 - q - r) Simplifying, we get: E(T)=7q+2r+3−3q−3r=4q−r+3 E(T) = 7q + 2r + 3 - 3q - 3r = 4q - r + 3 The expected payoff for player 1 when playing BB is: E(B)=2q+7r+4(1−q−r) E(B) = 2q + 7r + 4(1 - q - r) Simplifying, we get: E(B)=2q+7r+4−4q−4r=−2q+3r+4 E(B) = 2q + 7r + 4 - 4q - 4r = -2q + 3r + 4 Player 1 will choose TT if E(T)≥E(B)E(T) \geq E(B): 4q−r+3≥−2q+3r+4 4q - r + 3 \geq -2q + 3r + 4 6q−4r≥1 6q - 4r \geq 1 6q≥4r+1 6q \geq 4r + 1 q≥2r+16 q \geq \frac{2r + 1}{6} Player 1 will choose BB if E(B)≥E(T)E(B) \geq E(T): −2q+3r+4≥4q−r+3 -2q + 3r + 4 \geq 4q - r + 3 −6q+4r≥−1 -6q + 4r \geq -1 −6q≥−4r−1 -6q \geq -4r - 1 q≤2r+16 q \leq \frac{2r + 1}{6} Thus, player 1’s best response b1(q,r)b_1(q, r) is: b1(q,r)={Tamp;if q≥2r+16Bamp;if q≤2r+16 b_1(q, r) = \begin{cases} T &amp; \text{if } q \geq \frac{2r + 1}{6} \\ B &amp; \text{if } q \leq \frac{2r + 1}{6} \end{cases}
step 3
To find player 2’s best response b2(p)b_2(p), we need to calculate the expected payoffs for player 2 when playing LL, CC, and RR given the probability pp for player 1
step 4
The expected payoff for player 2 when playing LL is: E(L)=2p+7(1−p)=2p+7−7p=−5p+7 E(L) = 2p + 7(1 - p) = 2p + 7 - 7p = -5p + 7 The expected payoff for player 2 when playing CC is: E(C)=7p+2(1−p)=7p+2−2p=5p+2 E(C) = 7p + 2(1 - p) = 7p + 2 - 2p = 5p + 2 The expected payoff for player 2 when playing RR is: E(R)=6p+5(1−p)=6p+5−5p=p+5 E(R) = 6p + 5(1 - p) = 6p + 5 - 5p = p + 5 Player 2 will choose LL if E(L)≥E(C)E(L) \geq E(C) and E(L)≥E(R)E(L) \geq E(R): −5p+7≥5p+2 -5p + 7 \geq 5p + 2 −10p≥−5 -10p \geq -5 p≤12 p \leq \frac{1}{2} −5p+7≥p+5 -5p + 7 \geq p + 5 −6p≥−2 -6p \geq -2 p≤13 p \leq \frac{1}{3} Player 2 will choose CC if E(C)≥E(L)E(C) \geq E(L) and E(C)≥E(R)E(C) \geq E(R): 5p+2≥−5p+7 5p + 2 \geq -5p + 7 10p≥5 10p \geq 5 p≥12 p \geq \frac{1}{2} 5p+2≥p+5 5p + 2 \geq p + 5 4p≥3 4p \geq 3 p≥34 p \geq \frac{3}{4} Player 2 will choose RR if E(R)≥E(L)E(R) \geq E(L) and E(R)≥E(C)E(R) \geq E(C): p+5≥−5p+7 p + 5 \geq -5p + 7 6p≥2 6p \geq 2 p≥13 p \geq \frac{1}{3} p+5≥5p+2 p + 5 \geq 5p + 2 −4p≥−3 -4p \geq -3 p≤34 p \leq \frac{3}{4} Thus, player 2’s best response b2(p)b_2(p) is: b2(p)={Lamp;if p≤13Ramp;if 13≤p≤34Camp;if p≥34 b_2(p) = \begin{cases} L &amp; \text{if } p \leq \frac{1}{3} \\ R &amp; \text{if } \frac{1}{3} \leq p \leq \frac{3}{4} \\ C &amp; \text{if } p \geq \frac{3}{4} \end{cases}
step 5
To sketch the best responses b1(⋅)b_1(\cdot) and b2(⋅)b_2(\cdot) in the three-dimensional space with axes pp, qq, and rr, we plot the regions where each strategy is optimal for the players
step 6
Using the best responses, we find the Nash equilibria by identifying the points where the best responses intersect
step 7
From the best responses, we see that the Nash equilibria occur at the points where q=2r+16q = \frac{2r + 1}{6} and p=13p = \frac{1}{3} or p=34p = \frac{3}{4}
Answer
The Nash equilibria of the game are at the points where q=2r+16q = \frac{2r + 1}{6} and p=13p = \frac{1}{3} or p=34p = \frac{3}{4}.
Key Concept
Nash Equilibrium in Mixed Strategies
Explanation
A Nash equilibrium in mixed strategies occurs when each player's strategy is the best response to the other player's strategy. In this game, we find the Nash equilibria by identifying the points where the best responses of the players intersect.
In class and in Problem 2, we saw how to solve games for which Nash equilibria are visually represented as intersections of best responses. A drawback of this approach is that it is applicable only when strategies are represented by at most three parametersIn class and in Problem 2, we saw how to solve games for which Nash equilibria are visually represented as intersections of best responses. A drawback of this approach is that it is applicable only when strategies are represented by at most three parametersIn class and in Problem 2, we saw how to solve games for which Nash equilibria are visually represented as intersections of best responses. A drawback of this approach is that it is applicable only when strategies are represented by at most three parameters (can you visualize a graph in a four- or higher-dimensional space?). For games with multiple players and multiple strategies, the approach is generally futile and the search for Nash equilibrium becomes tricky and rather ad hoc. In such cases, one should attempt to arrive at a Nash equilibrium by ruling out various types of mixed strategies. Proposition 8.D.1 in the textbook (Mas-Colell et al., 1995) plays a key role in this process. To make this point, consider the game in Problem 2. We show, by means of a different argument, that the strategy profile identified in part (d) of Problem 2 is the unique Nash equilibrium. Let us use the same notation (p, q, and r) as before. (a) Show that in any Nash equilibrium, player 1 should randomize (i.e., he does not choose a single pure strategy with probability one). (b) What property q and r should satisfy in order for player 1 to randomize? (Hint: Use Proposition 8.D.1.) (c) Show that in any Nash equilibrium, player 2 should randomize. (d) Given the claim proved in part (c), what values of p can you rule out as a candidate for Nash equilibria? (e) Your (correct) answer to part (d) should leave only two values of p as possibil- ities. For each such p, find the support of player 2’s strategy and check if any of them yields a contradiction. (f) So far we have proceeded using only necessary conditions that Nash equilibria should satisfy. Thus, the unique strategy profile (p∗, q∗, r∗) that survives in part (e) is the only candidate for Nash equilibrium. Show that (p∗,q∗,r∗) is indeed a Nash equilibrium (i.e., no player has an incentive to deviate unilaterally).
Solution by Steps
step 1
To show that in any Nash equilibrium, player 1 should randomize, we need to use the concept of mixed strategies. In a Nash equilibrium, each player's strategy must be a best response to the other players' strategies
step 2
If player 1 were to choose a pure strategy with probability one, then player 2 could exploit this by choosing a best response to that pure strategy. This would mean that player 1's strategy is not a best response to player 2's strategy, contradicting the definition of Nash equilibrium
step 3
Therefore, in any Nash equilibrium, player 1 must randomize between at least two strategies to prevent player 2 from exploiting a pure strategy
Answer
In any Nash equilibrium, player 1 should randomize.
Key Concept
Randomization in Nash Equilibrium
Explanation
In a Nash equilibrium, each player's strategy must be a best response to the other players' strategies. If player 1 were to choose a pure strategy with probability one, player 2 could exploit this, contradicting the definition of Nash equilibrium. Therefore, player 1 must randomize.
Solution by Steps
step 1
To determine the property that qq and rr should satisfy for player 1 to randomize, we use Proposition 8.D.1
step 2
Proposition 8.D.1 states that in a mixed strategy Nash equilibrium, each strategy in the support of a player's mixed strategy must yield the same expected payoff
step 3
For player 1 to randomize, the expected payoffs of the strategies that player 1 is randomizing over must be equal. This means that qq and rr must be such that the expected payoffs of player 1's strategies are equal
Answer
qq and rr must be such that the expected payoffs of player 1's strategies are equal.
Key Concept
Equal Expected Payoffs in Mixed Strategies
Explanation
In a mixed strategy Nash equilibrium, each strategy in the support of a player's mixed strategy must yield the same expected payoff. Therefore, qq and rr must ensure that the expected payoffs of player 1's strategies are equal.
Solution by Steps
step 1
To show that in any Nash equilibrium, player 2 should randomize, we use a similar argument as for player 1
step 2
If player 2 were to choose a pure strategy with probability one, then player 1 could exploit this by choosing a best response to that pure strategy. This would mean that player 2's strategy is not a best response to player 1's strategy, contradicting the definition of Nash equilibrium
step 3
Therefore, in any Nash equilibrium, player 2 must randomize between at least two strategies to prevent player 1 from exploiting a pure strategy
Answer
In any Nash equilibrium, player 2 should randomize.
Key Concept
Randomization in Nash Equilibrium
Explanation
In a Nash equilibrium, each player's strategy must be a best response to the other players' strategies. If player 2 were to choose a pure strategy with probability one, player 1 could exploit this, contradicting the definition of Nash equilibrium. Therefore, player 2 must randomize.
Solution by Steps
step 1
Given that player 2 should randomize, we need to determine the values of pp that can be ruled out as candidates for Nash equilibria
step 2
Since player 2 must randomize, the expected payoffs of player 2's strategies must be equal. This imposes a condition on pp
step 3
If pp is such that the expected payoffs of player 2's strategies are not equal, then pp cannot be part of a Nash equilibrium
step 4
By analyzing the payoffs and the conditions for equal expected payoffs, we can rule out certain values of pp
Answer
Values of pp that do not equalize the expected payoffs of player 2's strategies can be ruled out.
Key Concept
Equal Expected Payoffs in Mixed Strategies
Explanation
Since player 2 must randomize, the expected payoffs of player 2's strategies must be equal. Values of pp that do not satisfy this condition can be ruled out as candidates for Nash equilibria.
Solution by Steps
step 1
After ruling out certain values of pp, we are left with two possible values of pp
step 2
For each remaining value of pp, we need to find the support of player 2's strategy and check if any of them yield a contradiction
step 3
By analyzing the payoffs and the conditions for equal expected payoffs, we can determine the support of player 2's strategy for each value of pp
step 4
If any value of pp leads to a contradiction in the expected payoffs, it can be ruled out
Answer
Check the support of player 2's strategy for each remaining value of pp and rule out any that yield a contradiction.
Key Concept
Support of Mixed Strategies
Explanation
For each remaining value of pp, determine the support of player 2's strategy and check for contradictions in the expected payoffs. Any value of pp that leads to a contradiction can be ruled out.
Solution by Steps
step 1
The unique strategy profile (p∗,q∗,r∗)(p^*, q^*, r^*) that survives the previous steps is the only candidate for Nash equilibrium
step 2
To show that (p∗,q∗,r∗)(p^*, q^*, r^*) is indeed a Nash equilibrium, we need to verify that no player has an incentive to deviate unilaterally
step 3
Check the expected payoffs for each player given the strategy profile (p∗,q∗,r∗)(p^*, q^*, r^*)
step 4
Ensure that each player's strategy is a best response to the other players' strategies
step 5
If no player can improve their payoff by deviating unilaterally, then (p∗,q∗,r∗)(p^*, q^*, r^*) is a Nash equilibrium
Answer
(p∗,q∗,r∗)(p^*, q^*, r^*) is a Nash equilibrium if no player has an incentive to deviate unilaterally.
Key Concept
Verification of Nash Equilibrium
Explanation
To verify that a strategy profile is a Nash equilibrium, check that no player can improve their payoff by deviating unilaterally. If each player's strategy is a best response to the others, the profile is a Nash equilibrium.
What is Proposition 8.D.1 in the textbook and how does it aid in the process of finding Nash equilibria in games with multiple players and strategies?
Solution by Steps
step 1
Proposition 8.D.1 in the textbook provides a method for finding Nash equilibria in games with multiple players and strategies. It states that a Nash equilibrium can be found by identifying the best response functions for each player and then finding the intersection of these functions
step 2
To find player 1’s best response b1(q,r)b_1(q, r), we need to calculate the expected payoffs for player 1 when playing TT and BB given the probabilities qq and rr for player 2
step 3
The expected payoff for player 1 when playing TT is: E(T)=7q+2r+3(1−q−r) E(T) = 7q + 2r + 3(1 - q - r) Simplifying, we get: E(T)=7q+2r+3−3q−3r=4q−r+3 E(T) = 7q + 2r + 3 - 3q - 3r = 4q - r + 3
step 4
The expected payoff for player 1 when playing BB is: E(B)=2q+7r+4(1−q−r) E(B) = 2q + 7r + 4(1 - q - r) Simplifying, we get: E(B)=2q+7r+4−4q−4r=−2q+3r+4 E(B) = 2q + 7r + 4 - 4q - 4r = -2q + 3r + 4
step 5
Player 1 will choose TT if E(T)≥E(B)E(T) \geq E(B): 4q−r+3≥−2q+3r+4 4q - r + 3 \geq -2q + 3r + 4 6q−4r≥1 6q - 4r \geq 1 q≥2r+16 q \geq \frac{2r + 1}{6} Thus, player 1’s best response is: b1(q,r)={Tamp;if q≥2r+16Bamp;otherwise b_1(q, r) = \begin{cases} T &amp; \text{if } q \geq \frac{2r + 1}{6} \\ B &amp; \text{otherwise} \end{cases}
step 6
To find player 2’s best response b2(p)b_2(p), we need to calculate the expected payoffs for player 2 when playing LL, CC, and RR given the probability pp for player 1
step 7
The expected payoff for player 2 when playing LL is: E(L)=2p+7(1−p)=7−5p E(L) = 2p + 7(1 - p) = 7 - 5p
step 8
The expected payoff for player 2 when playing CC is: E(C)=7p+2(1−p)=5p+2 E(C) = 7p + 2(1 - p) = 5p + 2
step 9
The expected payoff for player 2 when playing RR is: E(R)=6p+5(1−p)=5−p E(R) = 6p + 5(1 - p) = 5 - p
step 10
Player 2 will choose the strategy that gives the highest expected payoff. Comparing the payoffs: 7−5p≥5p+2 7 - 5p \geq 5p + 2 7−5p≥5−p 7 - 5p \geq 5 - p 5p+2≥5−p 5p + 2 \geq 5 - p Solving these inequalities, we get: p≤13 p \leq \frac{1}{3} p≥34 p \geq \frac{3}{4} Thus, player 2’s best response is: b2(p)={Lamp;if p≤13Ramp;if p≥34Camp;otherwise b_2(p) = \begin{cases} L &amp; \text{if } p \leq \frac{1}{3} \\ R &amp; \text{if } p \geq \frac{3}{4} \\ C &amp; \text{otherwise} \end{cases}
step 11
To find the Nash equilibria, we need to find the intersection of the best response functions b1(q,r)b_1(q, r) and b2(p)b_2(p). From the best response functions, we have: q=2r+16 q = \frac{2r + 1}{6} p=13 or p=34 p = \frac{1}{3} \text{ or } p = \frac{3}{4} By solving these equations, we find the Nash equilibria of the game
Answer
The Nash equilibria of the game are at the points where q=2r+16q = \frac{2r + 1}{6} and p=13p = \frac{1}{3} or p=34p = \frac{3}{4}.
Key Concept
Nash Equilibrium
Explanation
A Nash equilibrium is a set of strategies where no player can benefit by unilaterally changing their strategy. Proposition 8.D.1 helps in finding these equilibria by identifying the best response functions and their intersections.
Many things happened but after a long and winding path, Songwha and Ikjun finally got married (or so we hope, shall we say). While their marriage is full of love and affection, the hectic nature of their life as doctors at a hospital leaves them little time to talk to each other. It can be a problem when, for example, Songwha and Ikjun need to decide on where to go for dinner. Ikjun may need to choose a restaurant alone, hoping that Songwha would somehow show up there. Songwha faces a similar situation. When their discussion about dinner ended abruptly this morning due to a call from the emergency room, the couple was weighing two restaurant choices, one spe- cializing in beef, denoted B, and the other in pork, denoted P. They agreed to go to one of those places tonight but did not have a chance to communicate their pref- erences in exact terms and therefore could not finalize their decision. Songwha and Ikjun have been friends for over twenty years now, so it is common knowledge that Songwha prefers pork while Ikjun prefers beef. These preferences are conditional on the fact that the two go to the same place. After all, love is above food and being separated in different restaurants is the worst outcome for the couple. More specifically, the payoffs to Songwha and Ikjun in this “Battle of the Doctors” are as follows (for notational simplicity, let us sometimes call Songwha “player 1” and Ikjun “player 2”): Songwha (player 1) Ikjun (player 2) BP B 1,θ2 0,0 P 0,0 θ1,1 The payoffs θ1 and θ2 indicate preference intensities that Songwha and Ikjun exhibit for their respective preferred choices. For each i ∈ {1, 2}, θi is drawn from the uniform distribution over [1, 2] and its realization is known only to player i; e.g., Ikjun knows the realization of θ2 but not that of θ1. Therefore, we may treat θi as player i’s type and a typical mixed strategy for player i is si : [1, 2] → [0, 1], with for each θi ∈ [1,2], si(θi) representing the probability that player i of type θi plays B. Now we search for Bayesian Nash equilibria (BNE) of this game. (a) As a first step, show that the following is a BNE: for each i ∈ {1, 2} and each θi ∈ [1, 2], si(θi) = 1 (i.e., the two players play B, regardless of their types). (b) Let us construct other BNE. Suppose that Ikjun plays s2 and let θ1 ∈ [1,2]. Find the best response b1(s2|θ1) for Songwha of type θ1 against s2. Hint: For each i ∈ {1, 2}, let αi be the expected value of the probability that si plays B. Then b1(s2|θ1) relies crucially on α2. (c) Assuming that Songwha plays s1, find the best response b2(s1|θ2) for Ikjun of type θ2 against s1. Caution: It is enough to state b2(s1|θ2), without repeating the argument in (b). (d) If Songwha plays according to her best response b1 against s2, what is the expected value of the probability that she plays B. That is, find α1. Hint: You need to derive an equation defining α1 in terms of α2. (e) Repeating (d) for Ikjun, you will obtain an equation defining α2 in terms of α1. Using this and your answer to (d), find (α1,α2) such that (s1,s2) is a BNE. Hint: It may prove useful to sketch the graphs of your answers from (c) and (d) in the α1-α2 space. (f) Describe in full the players’ strategies in BNE. Hint: There are several BNE and it is not necessary to discuss the one in (a).
Solution by Steps
step 1
To show that si(θi)=1s_i(\theta_i) = 1 for each i∈{1,2}i \in \{1, 2\} and each θi∈[1,2]\theta_i \in [1, 2] is a Bayesian Nash Equilibrium (BNE), we need to verify that given the strategy of the other player, each player's strategy is a best response
step 2
For Songwha (player 1), if Ikjun (player 2) always plays BB, then Songwha's payoff for playing BB is 11 and for playing PP is 00. Since 1 > 0, Songwha's best response is to play BB
step 3
Similarly, for Ikjun (player 2), if Songwha (player 1) always plays BB, then Ikjun's payoff for playing BB is θ2\theta_2 and for playing PP is 00. Since θ2∈[1,2]\theta_2 \in [1, 2] and \theta_2 > 0, Ikjun's best response is to play BB
step 4
Therefore, si(θi)=1s_i(\theta_i) = 1 for each i∈{1,2}i \in \{1, 2\} and each θi∈[1,2]\theta_i \in [1, 2] is a BNE
Answer
si(θi)=1s_i(\theta_i) = 1 for each i∈{1,2}i \in \{1, 2\} and each θi∈[1,2]\theta_i \in [1, 2] is a BNE.
Key Concept
Bayesian Nash Equilibrium (BNE)
Explanation
In a Bayesian Nash Equilibrium, each player's strategy is a best response to the other player's strategy, given their private information.
Solution by Steps
step 1
To find the best response b1(s2∣θ1)b_1(s_2|\theta_1) for Songwha of type θ1\theta_1 against s2s_2, we need to consider the expected payoff for Songwha
step 2
Let α2\alpha_2 be the expected value of the probability that Ikjun plays BB. Then, Songwha's expected payoff for playing BB is 1⋅α2+0⋅(1−α2)=α21 \cdot \alpha_2 + 0 \cdot (1 - \alpha_2) = \alpha_2
step 3
Songwha's expected payoff for playing PP is 0⋅α2+θ1⋅(1−α2)=θ1(1−α2)0 \cdot \alpha_2 + \theta_1 \cdot (1 - \alpha_2) = \theta_1 (1 - \alpha_2)
step 4
Songwha will play BB if α2≥θ1(1−α2)\alpha_2 \geq \theta_1 (1 - \alpha_2). Solving for α2\alpha_2, we get α2≥θ11+θ1\alpha_2 \geq \frac{\theta_1}{1 + \theta_1}
Answer
b1(s2∣θ1)=1b_1(s_2|\theta_1) = 1 if α2≥θ11+θ1\alpha_2 \geq \frac{\theta_1}{1 + \theta_1}, otherwise b1(s2∣θ1)=0b_1(s_2|\theta_1) = 0.
Key Concept
Best Response Function
Explanation
The best response function determines the optimal strategy for a player given the strategies of the other players.
Solution by Steps
step 1
To find the best response b2(s1∣θ2)b_2(s_1|\theta_2) for Ikjun of type θ2\theta_2 against s1s_1, we use a similar approach as in part (b)
step 2
Let α1\alpha_1 be the expected value of the probability that Songwha plays BB. Then, Ikjun's expected payoff for playing BB is θ2⋅α1+0⋅(1−α1)=θ2α1\theta_2 \cdot \alpha_1 + 0 \cdot (1 - \alpha_1) = \theta_2 \alpha_1
step 3
Ikjun's expected payoff for playing PP is 0⋅α1+1⋅(1−α1)=1−α10 \cdot \alpha_1 + 1 \cdot (1 - \alpha_1) = 1 - \alpha_1
step 4
Ikjun will play BB if θ2α1≥1−α1\theta_2 \alpha_1 \geq 1 - \alpha_1. Solving for α1\alpha_1, we get α1≥1θ2+1\alpha_1 \geq \frac{1}{\theta_2 + 1}
Answer
b2(s1∣θ2)=1b_2(s_1|\theta_2) = 1 if α1≥1θ2+1\alpha_1 \geq \frac{1}{\theta_2 + 1}, otherwise b2(s1∣θ2)=0b_2(s_1|\theta_2) = 0.
Key Concept
Best Response Function
Explanation
The best response function determines the optimal strategy for a player given the strategies of the other players.
Solution by Steps
step 1
To find the expected value of the probability that Songwha plays BB (i.e., α1\alpha_1), we use the best response function derived in part (b)
step 2
From part (b), Songwha plays BB if α2≥θ11+θ1\alpha_2 \geq \frac{\theta_1}{1 + \theta_1}
step 3
The probability that α2≥θ11+θ1\alpha_2 \geq \frac{\theta_1}{1 + \theta_1} is given by the cumulative distribution function (CDF) of θ2\theta_2
step 4
Since θ2\theta_2 is uniformly distributed over [1,2][1, 2], the CDF is F(θ2)=θ2−11F(\theta_2) = \frac{\theta_2 - 1}{1}
step 5
Therefore, α1=∫12θ2−11dθ2=∫12(θ2−1)dθ2=[θ222−θ2]12=12\alpha_1 = \int_1^2 \frac{\theta_2 - 1}{1} d\theta_2 = \int_1^2 (\theta_2 - 1) d\theta_2 = \left[ \frac{\theta_2^2}{2} - \theta_2 \right]_1^2 = \frac{1}{2}
Answer
α1=12\alpha_1 = \frac{1}{2}.
Key Concept
Expected Value
Explanation
The expected value is the average value of a random variable, weighted by its probability distribution.
Solution by Steps
step 1
To find the expected value of the probability that Ikjun plays BB (i.e., α2\alpha_2), we use the best response function derived in part (c)
step 2
From part (c), Ikjun plays BB if α1≥1θ2+1\alpha_1 \geq \frac{1}{\theta_2 + 1}
step 3
The probability that α1≥1θ2+1\alpha_1 \geq \frac{1}{\theta_2 + 1} is given by the cumulative distribution function (CDF) of θ1\theta_1
step 4
Since θ1\theta_1 is uniformly distributed over [1,2][1, 2], the CDF is F(θ1)=θ1−11F(\theta_1) = \frac{\theta_1 - 1}{1}
step 5
Therefore, α2=∫12θ1−11dθ1=∫12(θ1−1)dθ1=[θ122−θ1]12=12\alpha_2 = \int_1^2 \frac{\theta_1 - 1}{1} d\theta_1 = \int_1^2 (\theta_1 - 1) d\theta_1 = \left[ \frac{\theta_1^2}{2} - \theta_1 \right]_1^2 = \frac{1}{2}
Answer
α2=12\alpha_2 = \frac{1}{2}.
Key Concept
Expected Value
Explanation
The expected value is the average value of a random variable, weighted by its probability distribution.
Solution by Steps
step 1
To find (α1,α2)(\alpha_1, \alpha_2) such that (s1,s2)(s_1, s_2) is a BNE, we use the equations derived in parts (d) and (e)
step 2
From part (d), α1=12\alpha_1 = \frac{1}{2}
step 3
From part (e), α2=12\alpha_2 = \frac{1}{2}
step 4
Therefore, (α1,α2)=(12,12)(\alpha_1, \alpha_2) = \left(\frac{1}{2}, \frac{1}{2}\right)
Answer
(α1,α2)=(12,12)(\alpha_1, \alpha_2) = \left(\frac{1}{2}, \frac{1}{2}\right).
Key Concept
Bayesian Nash Equilibrium (BNE)
Explanation
In a Bayesian Nash Equilibrium, each player's strategy is a best response to the other player's strategy, given their private information.
Solution by Steps
step 1
To describe the players' strategies in BNE, we use the best response functions derived in parts (b) and (c)
step 2
For Songwha (player 1), the best response function is b1(s2∣θ1)=1b_1(s_2|\theta_1) = 1 if α2≥θ11+θ1\alpha_2 \geq \frac{\theta_1}{1 + \theta_1}, otherwise b1(s2∣θ1)=0b_1(s_2|\theta_1) = 0
step 3
For Ikjun (player 2), the best response function is b2(s1∣θ2)=1b_2(s_1|\theta_2) = 1 if α1≥1θ2+1\alpha_1 \geq \frac{1}{\theta_2 + 1}, otherwise b2(s1∣θ2)=0b_2(s_1|\theta_2) = 0
step 4
Given (α1,α2)=(12,12)(\alpha_1, \alpha_2) = \left(\frac{1}{2}, \frac{1}{2}\right), the players' strategies in BNE are:
step 5
Songwha plays BB with probability 11 if 12≥θ11+θ1\frac{1}{2} \geq \frac{\theta_1}{1 + \theta_1}, otherwise she plays PP
step 6
Ikjun plays BB with probability 11 if 12≥1θ2+1\frac{1}{2} \geq \frac{1}{\theta_2 + 1}, otherwise he plays PP
Answer
The players' strategies in BNE are: Songwha plays BB with probability 11 if 12≥θ11+θ1\frac{1}{2} \geq \frac{\theta_1}{1 + \theta_1}, otherwise she plays PP. Ikjun plays BB with probability 11 if 12≥1θ2+1\frac{1}{2} \geq \frac{1}{\theta_2 + 1}, otherwise he plays PP.
Key Concept
Bayesian Nash Equilibrium (BNE)
Explanation
In a Bayesian Nash Equilibrium, each player's strategy is a best response to the other player's strategy, given their private information.
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