Compressing Mental Model Spaces and

Modeling Human Strategic Intent

Prashant Doshi University of Georgia, USA

http://thinc.cs.uga.edu

Yingke ChenDoctoral student

Yifeng ZengReader, Teesside Univ.

Previously: Assoc Prof., Aalborg Univ.

Hua MaoDoctoral student

Muthu ChandrasekaranDoctoral student

What is a mental behavioral model?

How large is the behavioral model space?

How large is the behavioral model space?

General definitionA mapping from the agent’s history of

observations to its actions

How large is the behavioral model space?

2H (Aj)Uncountably infinite

How large is the behavioral model space?

Let’s assume computable models

Countable

A very large portion of the model space is not computable!

Daniel DennettPhilosopher and Cognitive Scientist

Intentional stanceAscribe beliefs, preferences and intent to explain others’ actions

(analogous to theory of mind - ToM)

Organize the mental models

Intentional modelsSubintentional models

Organize the mental modelsIntentional models

E.g., POMDP = bj, Aj, Tj, j, Oj, Rj, OCj BDI, ToM

Subintentional models

Frame(may give rise to recursive modeling)

Organize the mental modelsIntentional models

E.g., POMDP = bj, Aj, Tj, j, Oj, Rj, OCj BDI, ToM

Subintentional modelsE.g., (Aj), finite state controller, plan

Frame

Finite model space grows as the interaction progresses

Growth in the model space

Other agent may receive any one of |j| observations

|Mj| |Mj||j| |Mj||j|2 ... |Mj||j|t

0 1 2 t

Growth in the model space

Exponential

Absolute continuity condition (ACC)

ACC1. Subjective distribution over histories2. True distribution over histories

ACC is a sufficient and necessary condition for Bayesian update of belief over models

How do we satisfy ACC?

Cautious beliefs (full prior) Grain of truth assumption

Prior with a grain of truth is sufficient but not necessary

General model space is large and grows exponentially as the interaction progresses

It would be great if we can compress this space!

No loss in value to the modelerFlexible loss in value for greater compression

LosslessLossy

Expansive usefulness of model space compression to many areas:

1. Sequential decision making (dt-planning) in multiagent settings

2. Bayesian plan recognition3. Games of imperfect information

1. Sequential decision making in multiagent settings

Interactive POMDP framework (Gmytrasiewicz&Doshi05)

Include models of the other agent in the state spaceUpdate beliefs over the physical state and models

General and domain-independent approach for compression

Establish equivalence relations that partition the model space and retain representative models from each equivalence class

Approach #1: Behavioral equivalence (Rathanasabapathy et al.06,Pynadath&Marsella07)

Intentional models whose complete solutions are identical are considered equivalent

Approach #1: Behavioral equivalence

Behaviorally minimal set of models

Lossless

Works when intentional models have differing frames

Approach #1: Behavioral equivalence

Multiagent tiger

Approach #1: Behavioral equivalence

Impact on dt-planning in multiagent settings

Multiagent tiger

Multiagent MM

Utilize model solutions (policy trees) for mitigating model growth

Approach #1: Behavioral equivalence

Model reps that are not BE may become BE next step onwards

Preemptively identify such models and do not update all of them

Redefine BE

Approach #2: -Behavioral equivalence(Zeng et al.11,12)

Intentional models whose partial depth-d solutions are identical and vectors of updated beliefs at the leaves of the partial trees

are identical are considered equivalent

Approach #2: Revisit BE(Zeng et al.11,12)

Sufficient but not necessary

Lossless if frames are identical

Approach #2: (,d)-Behavioral equivalence

Two models are (,d)-BE if their partial depth-d solutions are identical and vectors of updated beliefs at the leaves of the

partial trees differ by

Models are(0.33,1)-BE

Lossy

Approach #2: -Behavioral equivalence

Lemma (Boyen&Koller98): KL divergence between two distributions in a discrete Markov stochastic process reduces or remains the same after a transition, with the mixing rate acting as a discount factor

Mixing rate represents the minimal amount by which the posterior distributions agree with each other after one transition

Property of a problem and may be pre-computed

Given the mixing rate and a bound, , on the divergence between two belief vectors, lemma allows computing the depth, d, at which the bound is reached

Approach #2: -Behavioral equivalence

Compare two solutions up to depth d for equality

Discount factor F = 0.5

Multiagent Concert

Approach #2: -Behavioral equivalence

Impact on dt-planning in multiagent settings

Multiagent Concert

On a UAV reconnaissance problem in a 5x5 grid, allows the solution to scale to a 10 step look ahead in 20 minutes

What is the value of d when some problems exhibit F with a value of 0 or 1?

Approach #2: -Behavioral equivalence

F=1 implies that the KL divergence is 0 after one step: Set d = 1

F=0 implies that the KL divergence does not reduce: Arbitrarily set d to the horizon

Intentional or subintentional models whose predictions at time step t (action distributions)

are identical are considered equivalent at t

Approach #3: Action equivalence(Zeng et al.09,12)

Approach #3: Action equivalence

Lossy

Works when intentional models have differing frames

Approach #3: Action equivalence

Approach #3: Action equivalence

Impact on dt-planning in multiagent settings

Multiagent tigerAE bounds the model space at each time

step to the number of distinct actions

Intentional or subintentional models whose predictions at time step t influence the subject agent’s plan

identically are considered equivalent at t

Regardless of whether the other agent opened the left or right door,the tiger resets thereby affecting the agent’s plan identically

Approach #4: Influence equivalence(related to Witwicki&Durfee11)

Influence may be measured as the change in the subject agent’s belief due to the action

Approach #4: Influence equivalence

Group more models at time step t compared to AE

Lossy

Compression due to approximate equivalence may violate ACC

Regain ACC by appending a covering model to the compressed set of representatives

Open questions

N > 2 agents

Under what conditions could equivalent models belonging to different agents be

grouped together into an equivalence class?

Can we avoid solving models by using heuristics for identifying approximately

equivalent models?

Modeling Strategic Human Intent

Yifeng ZengReader, Teesside Univ.

Previously: Assoc Prof., Aalborg Univ.

Yingke ChenDoctoral student

Hua MaoDoctoral student

Muthu ChandrasekaranDoctoral student

Xia QuDoctoral student

Roi CerenDoctoral student

Matthew MeiselDoctoral student

Adam GoodieProfessor of Psychology, UGA

Computational modeling of human recursive thinking in sequential games

Computational modeling of probability judgment in stochastic games

Human strategic reasoning is generally hobbled by low levels of recursive thinking

(Stahl&Wilson95,Hedden&Zhang02,Camerer et al.04,Ficici&Pfeffer08)

(I think what you think that I think...)

You are Player I and II is human. Will you move or stay?

Move MoveMove

Stay Stay Stay

Payoff for I:Payoff for II:

31

13

24

42

I II I IIPlayer to move:

Less than 40% of the sample population performed the rational action!

Thinking about how others think (...) is hard in general contexts

Move MoveMove

Stay Stay Stay

Payoff for I:

(Payoff for II is 1 – decimal)

0.6 0.4 0.2 0.8

I II I IIPlayer to move:

About 70% of the sample population performed the rational action in this simpler and strictly competitive game

Simplicity, competitiveness and embedding the task in intuitive representations seem to facilitate

human reasoning (Flobbe et al.08, Meijering et al.11, Goodie et al.12)

3-stage game

Myopic opponents default to staying (level 0) while predictive opponents think about the player’s

decision (level 1)

Can we computationally model these strategic behaviors using process models?

Yes! Using a parameterized Interactive POMDP framework

Replace I-POMDP’s normative Bayesian belief update with Bayesian learning that underweights evidence, parameterized by

Notice that the achievement score increases as more games are played indicating learning of the opponent modelsLearning is slow and partial

Replace I-POMDP’s normative expected utility maximization with quantal response model that selects actions proportional to their utilities, parameterized by

Notice the presence of rationality errors in the participants’ choices (action is inconsistent with prediction) Errors appear to reduce with time

Underweighting evidence during learning and quantal response for

choice have prior psychological support

Use participants’ predictions of other’s action to learn and participants’ actions to learn

Use participants’ actions to learn both and Let vary linearly

Insights revealed by process modeling:1. Much evidence that participants did not make rote use of BI, instead

engaged in recursive thinking2. Rationality errors cannot be ignored when modeling human decision

making and they may vary3. Evidence that participants’ could be attributing surprising

observations of others’ actions to their rationality errors

Open questions:1. What is the impact on strategic thinking if action outcomes

are uncertain?2. Is there a damping effect on reasoning levels if participants

need to concomitantly think ahead in time

Suite of general and domain-independent approaches for compressing agent model

spaces based on equivalence

Computational modeling of human behavioral data pertaining to strategic thinking

Thank you for your time

2. Bayesian plan recognition under uncertainty

Plan recognition literature has paid scant attention to finding general ways of reducing the set of feasible

plans (Carberry, 01)

3. Games of imperfect information (Bayesian games)

Real-world applications often involve many player types Examples• Ad hoc coordination in a spontaneous team• Automated Poker player agent

3. Games of imperfect information (Bayesian games)

Real-world applications often involve many player types

Model space compression facilitates equilibrium computation