Feedback Design in Games

Feedback Design in Games with Ambiguity-Averse Players

• A study utilizes the notion of maxmin self-confirming equilibrium (MSCE) in designing information feedback for ambiguity-averse players.
• Coarse feedback influences strategic uncertainty, and modifies equilibrium strategies advantageously.
• Characterization of MSCE and equilibrium implications of coarse feedback happens in various games. Optimal design of feedback is to improve contributions in:
  • Generalized volunteer dilemmas
  • Public good games with strategic substitutes and complements
  • More general production functions
• Perfect and no feedback conditions are generally suboptimal.
• Some results are extended to 𝛼-maxmin preferences.

Introduction

• Nash equilibrium corresponds to a steady state of collective learning at the population level in n-player simultaneous-move games
• It is assumed whenever the game is played, each player i ∈ N playing the game is drawn from a large population of individuals assigned to the role of player i. Each individual plays the game only once
• With perfect feedback, individuals in each population i observe the aggregate behavior of individuals in every other population j ≠ i in ensuring their subjective theories about others' behavior are accurate at the steady state
• Nash equilibrium relies on players having sufficiently rich "archival access" to steady-state behavior
• Imperfect feedback can occur in many cases
• Individuals may observe only whether total or population-specific contributions in past plays reached a certain threshold in public goods problems
• In volunteer dilemmas individuals receive only coarse feedback about how many others volunteer.
• Uncertainty about others' behavior occurs due to Coarse feedback may, leading to influence on volunteering decisions.

Coarse Feedback and Equilibrium

• A whole range of subjective beliefs about the aggregate behavior in each population could be consistent with coarse feedback received by individuals
• Self-confirming equilibrium is the appropriate equilibrium concept, with the only restriction individuals' subjective beliefs is feedback consistency under the assumption of subjective expected utility maximization
• Nash equilibrium is always a self-confirming equilibrium, and the set of Nash and self-confirming equilibria coincide under perfect feedback
• Under perfect feedback, steady-state beliefs are accurate: population i observes that the share of individuals playing action aj in population j is sj(aj)
• Multiple steady-state frequencies of play consistent with the given feedback are possible since individuals do not perfectly observe the proportions of individuals playing in coarse feedback situations
• Each individual considers the worst-case scenario when deciding which strategy to play due to ambiguity aversion.
• Lehrer (2012) refers to steady-state behavior in such a setting as a partially specified equilibrium, while Battigalli et al. (2015, 2016) calls it a maxmin self-confirming equilibrium
• The study examines how equilibrium behavior of ambiguity-averse individuals is affected by coarse feedback, using a notion of maxmin self-confirming equilibrium adapted from Lehrer (2012) & Battigalli et al. (2015, 2016)
• The study examines the impact of coarse feedback from a design perspective
• Players do not know other' strategies; instead, the designer controls signals in standard Bayesian incentive problems and information design settings
• Strategic uncertainty is influenced by managing players' feedback, with the designer having no control over it
• Information design designs is all about designing "news access", feedback design centers on designing "archival access" or knowledge of steady-state behavior

Other Approaches

• The steady state of collective learning in games is affected using partial feedback as an instrument in other approaches
• The effect of coarse feedback is examined on self-confirming equilibrium bidding strategies and revenues in first-price auctions
• Revenue is studied in auction formats when bidders receive partial feedback about the distribution of bids submitted in earlier auctions using the notion of analogy-based expectation equilibrium
• Feedback is represented by analogy partitions of decision nodes from which players construct their subjective beliefs about the average behavior at these nodes
• Players form beliefs using maximum-entropy extrapolation after receiving partial feedback about payoff-irrelevant variables, others' behavior, and the state of nature in the games extended with payoff-irrelevant variables

Modeling

• Players rely solely on aggregate statistics from past interactions of other players.
• A designer can summarize and simplify this information to shape how players perceive others' behavior.
• Designers have access to past interaction data (regulator, manager, experimental economist, training coordinator, auction house, social networking service, or online gaming platform).

Designer Actions

• The designer reveals information to shape players' perception of others' behavior and influence their actions.
• The designer could add a column showing 1 if the sum of actions in a row exceeds a threshold k, and 0 otherwise, then remove all other columns in a binary action game example, where the designer has a large spreadsheet containing all past action profiles
• Public feedback sees all players receive the same spreadsheet
• Anonymous feedback sees the spreadsheet remain the same if the original columns are shuffled
• Separable feedback sees the designer reveal information about column j to player i independent of the columns k≠i, j
• Social networks have been explored to base feedback on by Lipnowski and Sadler (2019) in both simultaneous-move and dynamic games
• Designer aims to design feedback to induce favorable equilibrium behavior in order to increase voluntary contributions by exploiting individuals' aversion to ambiguity and providing only coarse feedback about individuals' behavior in past occurrences of the game Study Questions
• In past occurrences of the game is cooperation improved or deteriorated if players receive more or less ambiguous feedback?
• Are player's actions affected by players having aversion to ambiguity?

Game Setup

• Simultaneous-move games are examined with a finite number of players and an infinite set of strategies S¡ for each player j
• Set of strategies identified with the set of mixed actions is Finite games with a finite action set Aj for each player j
• Public good games with contributions represented by a number in the interval [0, 1], and finite games such as volunteer dilemma games are both considered
• Each player gets feedback about the strategies of his opponents, inducing a partition of the opponents' strategy sets, which can potentially depend on player i's strategy
• Maxmin self-confirming equilibrium (MSCE) aligns to the framework of Battigalli et al. (2016) which accommodates feedback that directly depends on the profile of mixed actions of opponents and, more generally, on a continuum of strategies
• Players are pessimistic and maximize their worst-case expected utility with respect to any strategy profile of the opponents that they consider possible given their feedback -Alpha-maxmin self-confirming equilibrium( a-MSCE) sees players consider best- and worst-case scenarios and maximize a weighted sum of best- and worst-case expected utilities to assess how robust some of the results are
• If players receive perfect feedback, they learn the equilibrium strategy profile perfectly, and MSCE and a-MSCE are equivalent to Nash equilibrium.
• Feedback functions characterized see to ensuring the existence of an MSCE
• The frameworks of partitially specfied equilibrium by Lehrer (201), and the existence result in Battigalli et al (2015, Theorem 2)
• The definition of feedback directly on mixed actions makes the framework allow for a broader range of statistics that players can learn about the behavior of other populations

Applied Examples

• A generalized class of volunteer dilemmas is considered, and the application of MSCE and alpha-MSCE
• Inducing MSCE outcomes with a higher number of volunteers requires allowing the coarse feedback
• Symmetric feedback could induce MSCE in which all players volunteer
• In games such as the standard volunteer dilemma game in which the public good is produced whenever at least one player volunteers, coarse feedback could be used to avoid coordination problems when the designer would like to incentivize only one player to volunteer A calss of continuous public good games studied
  • Strategic substitutes
  • strategic complements
  • More general forms of s-shaped production functions.
• Feedback can be characterized to increase cooperation and implies a higher average contribution than Nash equilibria
• The construction of coarse feedback can induce maximal contribution to the public good or n-times the Nash-equilibrium contribution
• For public good games with strategic substitutes, results are generalized to alpha-MSCE
• A class of games studied (strategic substitutes and negative externalities with linear best responses, such as Cournot oligopolies) led to provision of conditions and feedback functions that allow implementing the collusive outcome as an MSCE outcome

Model Components

• The simultaneous-move game G = N, S, u
  • N, the finite set of players, |N| = n
  • S set of strategy profiles
  • u utility functions
• For every player i, Si is a nonempty, compact, and convex subset of a Euclidean space, and ui is continuous and quasi-concave on Si The simultaneous-move game setup:
  • N finite set of players
  • Ai finite set of actions of player i
  • Si identified with the set of mixed actions of player i
• Player i's feedback is separable if it is received about each opponent's strategy Feedback Function
  • φi : S → Yi
  • Yi is a nonempty subset of a Euclidian space
• Strategy profile of players consistent with player i's feedback Σi Ω S−i
• Identification correspondence of player i - measures how player i deems his actions will impact steady state
• Player i gets perfect feedback: Σi(s) = { s-i }
• Player i gets no feedback: Σi(s) = S-i
• the feedback of player i satisfies own-strategy independence = Σi(si, s−i) = Σi
• A strategy profile s* is a maxmin self-confirming equilibrium (MSCE) if and only if inf uisi, s-i ³ inf uisi, si s2i[Si(s ) s2i[Si(s ) alpha-MSCE si arg max 3 inf uisi, si 41 a 5 sup uisi, si s,2S, s2i[Si(s ) s2i[Si(s ) - At perfect feedback the above reduces to NE
  • NE is pareto dominated by alpha - MSCE
  • MSCE - strategic ambiguity
  • Sufficient conditions to ensure alpha-MSCE
• Feedback function for player i- given by di 4 S ! Y i
• Each player knows actions of opponents at steady state

Results

Existence -

• Can be d-fined as NE of auxiliary game General observations
• Can consider worst payoff given feedback

Alpha considerations

• Can fail to exist, but if close then feedback quasi-concave Feedback type Open mapping function for a sufficient condition Then inverse correspondences are lower hemicontinuous Under assumptions, feedback is equivalent to NE of pure action

Other Types Equil

1. Battigalli The feedback is a function of a terminal mode of a finite game -Restricting, then simply definition on action profiles instead of mixed
2. Lehrer Similar concept to MSCE call partial Observing expected values random variables Then reduces to if pure, is limited Proposition 2 Under a condition then MSCE always exists