# lecture 7 Baum 1974

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```					  Baum, 1974

Generalized Matching Law
Describes basic matching law:
• P1/P1+P2 = R1/R1 + R2

• Revises to: P1/P2=R1/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) = a*log(R1/R2) + log b
–   P1/P2 = b(R1/R2)a
–   Where a = undermatching
–   B – bias
–   What is b and a? bias and undermatching
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
Alone
0.8

0.6

0.4
This is an example of almost

log (T1/T2)
0.2

0.0                                                                   perfect matching with little bias.
-0.2

-0.4
a = 0.99                 Why?
b = 0.02
-0.6
r2 = 0.99
-0.8
-0.8        -0.6    -0.4    -0.2    0.0    0.2   0.4   0.6   0.8

log (R1/R2)
Unpredicted Competitor
0.8

0.6                                                                  This is an example of
0.4

undermatching with some bias
log (T1/T2)

0.2

0.0
towards the RIGHT feeder.
Why?
-0.2

a = 0.59
-0.4
b = -.06
-0.6
r2 = 0.99
-0.8
-0.8     -0.6    -0.4    -0.2   0.0    0.2   0.4   0.6   0.8

log (R1/R2)
Predicted Competitor
0.8                                                                  This is an example of
0.6

0.4
overmatching with little bias.
log (T1/T2)

0.2
Why?
Is overmatching BETTER than
0.0

-0.2

a = 1.37
-0.4
b = 0.008                matching or undermatching?
-0.6
r2 = 0.96
-0.8
-0.8    -0.6    -0.4    -0.2    0.0    0.2   0.4   0.6   0.8
Why or why not?
log (R1/R2)
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?
Bias
•       Definition: magnitude of preference is shifted to one
reinforcer when there is apparent equality between the
rewards

•       Unaccounted for preference

•       Is experimenter’s 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
Four Sources of Bias
•   response bias

•   discrepancy between scheduled and obtained
reinforcement

•   qualitatively different reinforcers

•   qualitatively different reinforcement schedules
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
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 animal’s 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”
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
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.
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
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?
Now, can use it as a model to test
against!
• Let’s add a slightly different model
• Optimization models or Idea Free
Distribution
• Same idea, just with groups
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
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
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
the matching law.
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).
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.
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.
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
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.
•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.
Now let’s 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?
Baum and Kraft
• Examines this competition issue
• Might see REAL similarities between their
research and ours
• Be prepared to make some predictions