# Game Theory: Cheating Student and Professor ## 1. Identify Players and Strategies - **Players:** - Student - Professor - **Strategies:** - **Student:** Cheat (C), Not Cheat (NC) - **Professor:** Make Cheating Worthless (MW) (i.e., prepare a test where cheating does not help), Reward Honest (RH) students ## 2. Define Payoffs Let's assign reasonable payoffs based on the problem description: - **Student:** - Cheats and is not caught: High grade (e.g., 3) - Cheats and is caught: F and punishment (e.g., 0) - Does not cheat: Honest outcome, possibly rewarded (e.g., 2 if professor rewards, 1 otherwise) - **Professor:** - Wants to promote honesty and fairness. - If student cheats and succeeds: Unfair (e.g., 0) - If student cheats and is caught: Justice served (e.g., 2) - If student does not cheat: Prefers to reward (e.g., 3 if rewards honest, 2 otherwise) Payoff values are arbitrary but should reflect preferences. ## 3. Construct Payoff Matrix Let’s define the matrix as follows: | | **MW (Make cheating worthless)** | **RH (Reward Honest)** | |--------------|:-------------------------------:|:----------------------:| | **Cheat (C)**| (1, 2) | (3, 0) | | **Not Cheat (NC)**| (1, 2) | (2, 3) | - **(Student Payoff, Professor Payoff)** - **MW, Cheat:** Cheating doesn't help, so student gets only 1 (low grade), professor gets 2 (justice). - **MW, Not Cheat:** Student gets honest grade 1, professor gets 2. - **RH, Cheat:** Student cheats, gets high grade 3, professor is unhappy 0. - **RH, Not Cheat:** Student is honest, gets rewarded 2, professor is happy 3. ## 4. Analyze for Nash Equilibrium ### Step 1: Best Responses - **If Professor chooses MW:** - Student gets 1 whether cheats or not → **Student indifferent**. - **If Professor chooses RH:** - Cheat: Student gets 3 - Not Cheat: Student gets 2 → **Student prefers Cheat**. - **If Student chooses Cheat:** - MW: Professor gets 2 - RH: Professor gets 0 → **Professor prefers MW**. - **If Student chooses Not Cheat:** - MW: Professor gets 2 - RH: Professor gets 3 → **Professor prefers RH**. ### Step 2: Find Pure Strategy Nash Equilibria Check each cell: #### (C, MW): (1,2) - Student: Indifferent (gets 1 in MW whether cheats or not) - Professor: If student is cheating, MW is better (2 vs 0). - **Possible equilibrium, but student could also not cheat with same payoff.** #### (C, RH): (3,0) - Professor would switch to MW (gets 2 instead of 0). **Not an equilibrium.** #### (NC, MW): (1,2) - Same as (C, MW). Student could switch to cheating with no loss. #### (NC, RH): (2,3) - Student could switch to cheating and get 3 (since in RH, cheating is better). **Not an equilibrium.** ### Step 3: Mixed Strategy Equilibrium Let: - Student cheats with probability \( p \) - Professor chooses RH with probability \( q \) #### Set up equations - **Student's expected payoff:** \( U_S = q \cdot [p \cdot 3 + (1-p) \cdot 2] + (1-q) \cdot [p \cdot 1 + (1-p) \cdot 1] \) - **Professor's expected payoff:** \( U_P = p \cdot [q \cdot 0 + (1-q) \cdot 2] + (1-p) \cdot [q \cdot 3 + (1-q) \cdot 2] \) But for mixed equilibrium, each player must be indifferent between their own strategies. #### For Student: Indifference between Cheat and Not Cheat - **Cheat:** \( q \cdot 3 + (1-q) \cdot 1 = 3q + 1 - q = 2q + 1 \) - **Not Cheat:** \( q \cdot 2 + (1-q) \cdot 1 = 2q + 1 - q = q + 1 \) Set equal: \[ 2q + 1 = q + 1 \implies q = 0 \] So, if professor never rewards honest, student is indifferent. #### For Professor: Indifference between MW and RH - **MW:** \( p \cdot 2 + (1-p) \cdot 2 = 2 \) - **RH:** \( p \cdot 0 + (1-p) \cdot 3 = 0 + 3(1-p) = 3 - 3p \) Set equal: \[ 2 = 3 - 3p \implies 3p = 1 \implies p = \frac{1}{3} \] So, if student cheats with probability \( \frac{1}{3} \), professor is indifferent. ## 5. Final Answers ### **Payoff Matrix** | | **MW** | **RH** | |--------------|:-----------:|:-----------:| | **Cheat** | (1, 2) | (3, 0) | | **Not Cheat**| (1, 2) | (2, 3) | ### **Nash Equilibrium** - **Pure strategy:** None (other than both playing MW/any by student, which is not strict). - **Mixed strategy:** - **Student cheats with probability** \( p = \frac{1}{3} \), does not cheat with probability \( \frac{2}{3} \). - **Professor chooses MW with probability 1**, RH with probability 0. **Interpretation:** The professor's best defense is always to make cheating worthless (MW), which makes the student indifferent between cheating and not cheating (since payoffs are the same). If the professor sometimes rewards honest students, the student would cheat more often. Thus, the equilibrium is that the professor always picks MW, and the student is indifferent. --- ## **Summary Table** | | MW (Make Cheating Worthless) | RH (Reward Honest) | |------------|:----------------------------:|:------------------:| | Cheat | (1, 2) | (3, 0) | | Not Cheat | (1, 2) | (2, 3) | **Equilibrium:** - Professor: Always chooses MW - Student: Indifferent, any mix of cheating or not cheating yields same payoff --- **Key Point:** The most stable outcome is for the professor to always design tests where cheating does not help. This eliminates the incentive for students to cheat.