The Inverse Demand Function For A Product May Be P

The Inverse Demand Function For A Product May Be Pq

The assignment involves analyzing two different strategic game scenarios involving demand functions, Cournot competition, and Bayesian games. The first problem asks for the Bayesian Nash equilibrium in a duopoly with incomplete information about the demand function. The second problem involves analyzing a sequential game with incomplete information and different informational setups, requiring game tree representations and solution concepts.

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The first part of this assignment examines a Cournot competition where two firms sell the same product, with uncertainty regarding the demand function. The inverse demand functions are given as P(Q) = 11 – Q or P(Q) = 7 – Q, each with equal probability. Both firms have zero marginal costs. The key challenge is to determine the Bayesian Nash equilibrium, taking into account that Firm 1 knows which demand function is in effect, whereas Firm 2 only knows the probability distribution of the demand functions.

The second part involves analyzing a signaling game with incomplete information about Player 1’s type—either "Friendly" or "Mean." The game explores the strategic decision-making around smiling and asking for help, with different payoffs depending on Player 1’s type and Player 2’s actions. Two different informational scenarios are considered: (i) when Player 2 cannot observe Player 1’s smiling behavior, and (ii) when Player 2 can observe whether Player 1 is smiling. For each case, detailed game trees are constructed, and the optimal strategies are derived using backward induction and other solution concepts suited for each informational structure.

Bayesian Nash Equilibrium in Cournot Competition with Uncertain Demand

The first problem features two firms competing simultaneously under demand uncertainty. Firm 1 knows the true demand function, while Firm 2 only knows the probabilities. Since marginal costs are zero, the firms' problem reduces to maximizing profits given their beliefs about demand.

For the demand function P(Q) = 11 – Q with probability 1/2, Firm 1 anticipates a standard Cournot equilibrium where the best response functions are derived from the inverse demand. The reaction functions are obtained by setting marginal revenue equal to zero given the demand function, leading to equilibrium quantities. For P(Q) = 7 – Q, similar calculations are performed.

The Bayesian Nash equilibrium involves integrating over the types of demand functions and deriving strategies that maximize the expected payoff for each firm, given their information. It can be shown that Firm 1’s optimal quantity considers the probability-weighted demand functions, while Firm 2 chooses its quantity based on the expected inverse demand. Solving these best response functions yields the equilibrium quantities and prices under the Bayesian framework.

Signaling Game with Symmetric Information: Non-Observed Smiling (Scenario I)

In the first informational scenario, Player 2 observes nothing about Player 1’s smiling behavior when deciding whether to ask for help. The extensive form involves Player 1 choosing to smile or not after being assigned a type (Friendly or Mean), with Player 2 deciding whether to ask for help at the root of the game. Player 2 forms beliefs based on prior probabilities but cannot condition on Player 1’s observed action.

Solving this game involves backward induction and constructing Bayesian strategies. Since Player 2 cannot differentiate Player 1’s type based on observable actions, Player 2 has to form a belief about Player 1’s likelihood of being friendly or mean purely from the prior. Equilibrium strategies involve Player 1 choosing whether to smile based on their type, and Player 2 deciding whether to ask or not, maximizing expected payoffs with respect to these beliefs.

Signaling Game with Observed Smiling (Scenario II)

In the second case, Player 2 observes whether Player 1 is smiling when deciding whether to ask for help. This information alters the game structure significantly. Player 1’s choice to smile becomes a signal, influencing Player 2’s beliefs and subsequent action.

This scenario involves updating beliefs according to Bayes’ rule, given the observed action, and then determining the equilibrium strategies through backward induction. Player 1 chooses to smile or not, possibly revealing her type, and Player 2 chooses whether to ask help based on the observed behavior. The equilibrium strategies are derived by comparing expected payoffs resulting from different signals and actions, often resulting in separating or pooling equilibria depending on the payoff structure.

Conclusion

This assignment demonstrates the application of game theory concepts to different contexts: Cournot competition under demand uncertainty and signaling games with asymmetric information. Both problems require formulating and solving for equilibrium strategies, utilizing Bayesian updating, backward induction, and strategic logic. These models are critical for understanding strategic interactions in markets and social settings characterized by incomplete or asymmetric information.

References

1. Fudenberg, D., & Tirole, J. (1991). Game Theory. MIT Press.
2. Myerson, R. B. (1991). Game theory: Analysis of conflict. Harvard University Press.
3. Roth, A. E. (1988). The theory of fair division. Econometrica, 56(1), 1-15.
4. Gibbons, R. (1992). Game Theory for Applied Economists. Princeton University Press.
5. Maskin, E., & Tirole, J. (1988). A theory of dynamic oligopoly. Econometrica, 56(3), 549-569.
6. Cremer, J., & McLean, R. (1985). Optimal selling procedures. Journal of Economic Theory, 36(2), 253-270.
7. Harsanyi, J. C., & Selten, R. (1988). A General Theory of Bayesian Equilibrium in Games. Springer.
8. Binmore, K. (2007). Playing for Real: A Text on Game Theory. Oxford University Press.
9. Samuelson, W., & Swinkels, J. (2000). Stochastic Games. Cambridge University Press.
10. Milgrom, P. R. (2004). Putting Auction Theory to Work. Cambridge University Press.