Game Theory: Continuity in equilibrium profits?

Question

Consider 2 agents $A_i$. $A1$ moves before agent $A2$. Each of their utility functions is continuous in each agents' decision $0<s_i\in \mathbb{R}$ and a parameter $x$. Additionally, each agent's utility function has a unique interior maximum in its own decision $s_i$.

The agents' have lexicographic prefeences (primarily needed so that the other agents can anticipate strategies): $A_1$'s maximize its own utility, if two or more strategies maximize its own strategy it chooses among these the one that is best for $A_2$. $A_2$'s maximizes its own utility, if two or more strategies maximize its own strategy it chooses among these the one that is best for $A_1$.

Statement one: This game has at least one SPNE, but in general multiple. In case of multiple SPNE these will have the same utility profiles, that is, yield each agent the same utility.

Statement two: The agent's utility profiles are continuous in $x$ (Remember, each of their utilities was continuous in $x$.) My reasoning for statement two is that an incremental change in x implies that the ne NE will be very close to one of the NE before the change in $x$. And as these NE had the same utility profile, the change in $x$ will only have an incremental effect on the utility profiles.

Do you agree with the two statements and do you perhaps have a more solid argumentation (partcularly for the second).

Answer 1

Let me first try to answer Statement 2:

The agent's utility profiles are continuous in x

Let's try to flesh out the problem. Let $S_1$ and $S_2$ be compact, convex, strategy sets of the players (say subsets of $\mathbb{R}^n$) (for simplicity) . Let $u_1(s_1, s_2, x)$ and $u_2(s_1, s_2,x)$ be the continuous utility functions of the two players.

Let's solve for the SPNE. At stage 2, player 2 maximizes her utility given the action $s_1$ of player 1. This gives: $$ v_2(s_1, x) = \max_{s_2 \in S_2} u_2(s_1, s_2, x)\\ s_2(s_1, x) = \arg\max_{s_2 \in S_2} u_2(s_1, s_2, x). $$

• The problem is well defined due to the theorem of the maximum.
• The function $v_2(s_1, x)$ is continuous (Berge's maximisation theorem).
• The best response $s_2(s_1, x)$ is only guaranteed to be an upper hemi-continuous, non-empty correspondence.
• If $s_2(s_1, x)$ is single valued, then it is continuous. However single valuedness will depend on the functional form of $u_2(s_1, s_2, x)$ (e.g. strict quasi-concavity) and therefore does not follow from the assumptions so far.

Let $\tilde s_2(s_1, x)$ be a selection of $s_2(s_1, x)$. In stage 1, Player 1's optimisation problem is: $$ v_1^\ast(x) = \max_{s_1 \in S_1} u_1(s_1, \tilde s_2(s_1, x), x).\\ s_1^\ast(x) = \arg\max_{s_1 \in S_1} u_1(s_1, \tilde s_2(s_1, x), x). $$

• If the selection $\tilde s_2(s_1, x)$ of Player 2 is the optimal choice from $s_2(s_1, x)$ for Player 1, then the objective function appears to be upper semi-continuous, (but not, in general, continuous). Due to upper semi-continuity of the objective function, the maximisation problem is well defined.
• Although $v_1^\ast$ will be (I think) upper semi-continuous, it is not necessarily continuous. Anyway, the best response correspondence $s_1^\ast(x)$ does not need to be continuous.
• If $s_2(s_1, x)$ was single valued, then it is continuous, which means that $v_1^\ast(x)$ is also continuous. If, in addition $s_1^\ast(x)$ is also single valued, then it will also be continuous.

Let $\tilde s_1(x)$ be a selection from the best response $s_1^\ast(x)$ from player 1. Then, we can define: $$ v_2^\ast(x) = v_2(\tilde s_1(x), x),\\ s_2^\ast(x) = \tilde s_2(\tilde s_1(x), x). $$

• In general, neither $v_2^\ast(x)$ nor $s_2^\ast(x)$ need to be continuous.
• If both $s_1^\ast(x)$ and $s_2(s_1, x)$ are single valued, then $v_2^\ast(x)$ and $s_2^\ast(x)$ will both be continuous.

It thus appears that Statement 2 holds if both $s_2(s_1, x)$ and $s_1^\ast(x)$ are single valued.

• Single valuedness of $s_2(s_1, x)$ holds if (for example) $u_2(s_1, s_2, x)$ is strictly quasi-concave in $s_2$
• In turn, single valuedness of $s_1^\ast(x)$ holds if $u_1(s_1, \tilde s_2(s_1, x), s_1)$ is strictly quasi-concave in $s_1$. Notice that strict quasi-concavity of $u_1(s_1, s_2(s_1, x), x)$ is a strong assumption as it puts shape restrictions on the best response $s_2(s_1, x)$ (and not only on the function $u_1$ itself).

I hope this answers Statement 2.

For statement 1:

This game has at least one SPNE, but in general multiple. In case of multiple SPNE these will have the same utility profiles, that is, yield each agent the same utility.

• As shown above, the existence of a SPNE can be shown using only continuity of $u_1$ and $u_2$ and compactness of the strategy sets, and under the assumption that $\tilde s_2(s_1, x)$ selects the best option for Player 1.
• Different SPNE will in general not give the same utility. Assume that at stage 2, Player 2 is indifferent between more than one optimal strategy, so $s_2(s_1, x)$ is multivalued (for example, assume that $u_2(s_1, s_2, x)$ is the constant function, so Player 2 is indifferent between every strategy in $S_2$). If different selections $\tilde s_2(s_1, x)$ from $s_2(s_1, x)$, provide different utilities for player 1, then it is possible that $v_1^\ast$ takes different values depending on the choice of player 2 in stage 2 of the game. So different SPNE may give different utilities.

Answer 2

This game need not have any equilibrium as defined here with the lexicographic structure. Consider a situation where the payoff depends only on the action $x$ of player $1$ with $u_1(x)=\sin x$ and $u_2(x)=x$. There are infinitely many maxima of $u_1$, but no largest maximizer. So one cannot select a best choice for player $2$ among those that are optimal for player $1$.