Baum, 1974

1
Baum, 1974

• Generalized Matching Law

2
Describes basic matching law

• P1/P1P2 R1/R1 R2
• Revises to P1/P2R1/R2
• Notes that Staddon (1968) found can log it out to
  get straight lines
• Also adds two parameters k and a (we will use b
  and a)
• New version Log(P1/P2) alog(R1/R2) log b
• P1/P2 b(R1/R2)a
• Where a undermatching
• B bias
• What is b and a? bias and undermatching

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What is Undermatching?

• Fantino, Squires, Delbruck and Peterson (1972)
  defined
• Any preference less extreme than the matching
  relation would predict
• Systematic deviation from the matching relation
  for preferences toward both alternatives, in the
  direction of indifference
• What would be indifference? What value of the
  slope of the line?
• What would we call it when the slope of a of the
  line fitted according to equation is LESS THAN
  one?
• Greater than one?
• in a sense, is a discrimination or sensitivity
  model tells us how sensitive the animal is to
  changes in the (rate) of reward between the two
  alternatives

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This is an example of almost perfect matching
with little bias. Why?
This is an example of undermatching with some
bias towards the RIGHT feeder. Why?
This is an example of overmatching with little
bias. Why? Is overmatching BETTER than matching
or undermatching? Why or why not?
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Factors affecting the a or undermatching
parameter

• Discriminability between the stimuli signaling
  the two schedules
• Discrminability between the two rates of
  reinforcers
• Component duration
• COD and COD duration
• Deprivation level
• Others?

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Bias

• Definition magnitude of preference is shifted to
  one reinforcer when there is apparent equality
  between the rewards
• Unaccounted for preference
• Is experimenters failure to make both
  alternatives equal!
• Calculated using the intercept of the line
• Positive bias is a preference for R1
• Negative bias is a preference for R2

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Four Sources of Bias

• response bias
• discrepancy between scheduled and obtained
  reinforcement
• qualitatively different reinforcers
• qualitatively different reinforcement schedules

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Response bias

• Difficulty of making response one response key
  harder to push than other
• Qualitatively different reinforcers Spam vs.
  cream brulee
• Color
• Side of box, etc

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Difference between scheduled and obtained rate of
reinforcement

• Animal pauses, lowers obtained reinforcement even
  though programmed at higher rate (delivery
  dependent on responding!)
• Thus matching law applies only to obtained
  reward,
• if large discrepancies between obtained and
  scheduled, must use obtained to see animals
  preference
• If use wrong version of R1 and R2, can created
  LARGE bias rather than changes in reward
  sensitivity
• Other data suggests that this may not be true
• animals attend to programmed or scheduled reward
  in social situations
• May react because they are not getting what they
  expected or thought they were supposed to get

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Qualitatively Different Rewards

• Matching law only takes into consideration the
  rate of reward
• If qualitatively different, must add this in
• So P1/P2 V1/V2(R1/R2)a
• Must add in additional factor for qualitative
  differences
• Interestingly, can get u-shaped functions rather
  than hyperbolas this way move to economic models
  that allow for U-shaped rather than hyperbolic
  functions.

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Qualitatively different reinforcement schedules

• Use of VI versus VR
• Animal should show exclusive choice for VR, or
  minimal responding to VI
• Can control response rate, but not time
• Not match in typical sense, but is still
  optimizing

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So, does the matching law work?

• Matching holds up well under mathematical and
  data tests
• some limitations for model
• tells us about sensitivity to reward and bias
• now where would social interactions fit into
  this?

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Now, can use it as a model to test against!

• Lets add a slightly different model
• Optimization models or Idea Free Distribution
• Same idea, just with groups

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Optimal Foraging/Ideal Free Distribution

• A model of optimal foraging which describes the
  relative distribution of foraging animals between
  patches differing in resource density
• In its simplest form, the ideal free distribution
  predicts that the relative number of animals in
  each of two patches (N1 and N2 ) will be related
  to the relative resource density of the two
  patches (A1 and A2).
• N1/N2 A1/A2

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Similarity to the Matching Law

• It its logarithmic form, the ideal free
  distribution is described by
• log (N1/N2) a log (A1/A2)
  log b
• a represents the degree of sensitivity of the
  group behavior to differences in resource
  distribution.
• b represents a greater (or lesser) number of
  animals than expected in a patch for reasons
  unrelated to resource distribution. (e.g.,
  predation danger between patches).
• Equation 1 and Equation 2 are obviously quite
  similar, and a number of recent authors have in
  fact explored the similarities between the models

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Generalized Matching Law Ideal Free
Distribution?

• The competition dimension is a particularly
  critical difference between the models because
  competition drives predictions of ideal free
  distribution.
• Despite the formal similarities, some important
  differences
• the matching law describes the behavior of a
  single animal exposed to two sources of
  reinforcement,
• while the ideal free distribution describes the
  distribution of multiple animals between two
  resource patches.
• The presence of multiple animals introduces a
  dimension of competition into the ideal free
  distribution that is not found in the matching
  law.

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Differing Predictions

• Consider a foraging environment with two patches
  producing resources at different rates.
• When a single animal is present, foraging at the
  high-rate patch is clearly the better strategy
  (assuming low rates of changeover between
  patches).
• Changes when multiple animals present
• increase in the number of animals present in a
  patch increases competition for resources,
• This, in turn, decreases the rate at which an
  individual animals acquires the resource
  (individual capture rate).

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Individual vs. group

• Under such conditions, an individual can increase
  its capture rate by adopting a contrarian
  strategy and foraging in the patch with fewer
  resources but fewer competitors as well.
• The system will be at equilibrium when capture
  rates for individuals at each patch are equal.
• Assuming that all animals are equally good
  competitors, this will occur when the relative
  number of animals in a patch equals (or matches)
  the relative rate at which the patch produces
  resources.

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Interesting asymmetry between the models.

• A group of animals each individually following
  the matching law would also, over the long run,
  adhere to the ideal free distribution.
• The opposite is not true, though. All
  individuals may not match.
• It is possible for a group of animals to follow
  the ideal free distribution but for few or none
  of the individual animals to adhere to the
  matching law.
• This would occur, for example, if animals were
  distributed between patches in proportion to
  resource density but did move between patches.

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Competition and Matching?

• Competition clearly is an important difference
  between the models.
• Competition itself has been widely studied,
  especially in relation to the ideal free
  distribution
• However, competition has rarely been studied
  explicitly in the context of the matching law.
• These questions might be addressed by introducing
  an element of competition into a matching
  paradigm.

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How might sensitivity change?

• Introduction of a competitor might serve as a
  distraction resulting in a decreased
  sensitivity to reinforcement (see Baum, 1974).
• Alternatively, introduction of a competitor might
  increase the importance of sensitivity to
  reinforcement because presence of a competitor
  reduces the individual capture rate.

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How might bias change?

• It is quite possible that bias would not change
  at all.
• Alternatively, the presence of a competitor might
  cause a subject to remain in one or the other
  resource patch independent of reinforcement rate,
  which would be reflected in an increased bias.

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Now lets look at OUR rats

• What kind of data are we collecting?
• Matching law data
• IDF data
• Can we determine individual rat
• A parameter, sensitivity to reward or Matching
• Bias
• Can we determine ideal free distribution for the
  group?
• What kinds of differences should we see?
• Why?

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Reading for Next WeekBaum and Kraft

• Examines this competition issue
• Might see REAL similarities between their
  research and ours
• Be prepared to make some predictions regarding
  your rats