Q$3$: ERC Calculations $(1$ point$)$

In the ERC model, each player$$s utility function Ui depends on their own monetary payoff xi and the distribution of payoffs. The utility function is:

Ui$=$xi$$i$($xixi$+$xj$1$n$)2$

where:

xi is the monetary payoff of Player i

xj is the monetary payoff of the other player $($j$)$

i is Player i$\text{'}$s inequality aversion parameter

n represents the number of participants in the game $(2$ in all cases below$)$

This function implies that Player i$\text{'}$s utility decreases as the inequality between their payoff and the other player$$s payoff increases, weighted by their inequality aversion parameter i$.$

Consider an ultimatum game where Player i $($the proposer$)$ decides how to split $$100$ between themselves and Player j $($the responder$).$ Player i has an inequality aversion parameter i$=25,$ and Player j has an inequality aversion parameter j$=1220.$ Using the ERC model, calculate the minimum share xj $($where xj represents the amount offered to Player j$)$ that Player i needs to offer for Player j to accept the proposal. Explain why this share is necessary by showing Player j$$s utility calculations if xj is lower than this minimum.

In a dictator game, Player i can split $$50$ however they wish between themselves and Player j$.$ Player i has an inequality aversion parameter i$=15.$ Calculate the share xj that Player i will choose to give to Player j$.$

Consider an ultimatum game where Player i $($the proposer$)$ decides how to split $$5$ between themselves and Player j $($the responder$).$ Player i decides to propose a split $($xi$=$$$3\mathrm{.}50,$xj$=$$$1\mathrm{.}50).$ Calculate the maximum value of j that would lead Player j to feel satisfied enough with the offer that they would accept it$.$

In a dictator game, Player i can split $$50$ however they wish between themselves and Player j$.$ Player i has an inequality aversion parameter i$=15$ and Player j has an inequality aversion parameter j$=35.$ Calculate the share xj that Player i will choose to give to Player j if Player i has a utility function of the form below:Ui$($xi$,$xj$)=$xi$$i$($xixi$+$xj$1$n$)2+$xj$$j$($xjxi$+$xj$1$n$)2$