# WRITING 151: GAMES WE PLAY, PBE, 4

[15MIN]

When type is private information, the games become more complicated. This is because, for an equilibrium to exist, each player must be able to correctly predict others’ actions.

In a Bayesian game, player 1 can be of different types (assigned by mother Nature). He can observe his type, but player 2 knows only the type distribution. The issue becomes trickier when player 2 has to move in the ensuring information set: even after observing player 1’s action, player 2 is still uncertain about player 1’s type. Without further information, player 2 cannot decide his best response.

This technical difficulty is resolved by introducing the notion of belief. Specifically, for a given Bayesian Nash equilibrium (BNE), at each information set, each player must hold the correct belief that is consistent with the BNE; he then plays the best response to the belief. At the information set reached with positive probability, the belief is updated according to the Bayes’ rule. Off of the equilibrium path, however, the belief can take arbitrary form.

Formally, a Bayesian Nash equilibrium and a consistent belief constitute a perfect Bayesian equilibrium (PBE). Hence, PBE has four requirements: 1) every player holds a belief, 2) on the equilibrium path, the belief follows Bayes’ rule; 3) off the equilibrium path, the belief is arbitrary; 4) the BNE is each player’s best response to that belief.

In a sense, the belief is an intermediate concept that ensures the sequential rationality of BNE (Bayes’ rule and best response). The idea is similar to that of mean field theory.

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# DAY 150, 10PTS, 2.14S

[ROUTINE]
RUN, 2.3MILES, 9MPH, 15MIN;
SWIM: 1K;
LEARN: LUENBERGER;

[WORK]
AO: WRITE, PELICAN, 3HRS;
TQ2: READ, 1HR;

[DISCIPLINE]
CALL COX TO FIX THE TV ONLINE PROBLEM (WI-FI WIRELESS CONNECTION IN TV’S NETWORK SETTING), 15MIN;
CLEAN UP MAILS, PAY BILLS, 30MIN;

[UNHAPPY MOMENT]
BIG FIGHT WITH DAD; BUT I WON’T BUDGE.

# WRITING 140: THE MAIN IDEAS OF AO

[7:15pm-7.35pm, 20min]

AO is an interesting game with following features. M produces a product. He can sells either through R or directly to the end market. The market price is linear in the total quantity supplied. M is less efficient in selling; he incurs additional selling cost. The selling cost can be high or low, which is M’s private information. R only knows its prior distribution $\mu$. Finally, both M and R are risk-neutral profit-maximizing.

The game plays out as follows. M first observes his selling cost, and then he decides the wholesale price. Afterwards, R places the order, followed by R deciding his direct selling quantity. Finally, M produces, delivers, and sells the product through both channels.

Three forces are in play: DM in the retail channel, signaling because of IA, and the ENC competition between the two channels. Without IA, channel competition may reduce DM, i.e., competition is beneficial. However, when IA is in place, the outcome is less conclusive. In particular, if IA is on the R’s side, it actually exacerbates DM, reducing the benefit from channel competition.

AO argues that, however, when IA arises from M’s side, it works to further reduce DM, and hence a beneficial factor. As a result, the answer to whether IA is beneficial or harmful is more nuanced: it depends critically on the nature of IA, i.e., on the M or R’s side. It also depends on the relative selling costs. By spelling out these intricacies, our paper brings a step closer to the better understanding of pros and cons of IA and ENC competition in channel management.

[SUNSET, NEWPORT BEACH, JUNE, 2015]

# Newport Beach Public Library

Modern, clean, and quiet: I like it.