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Class Business

Class Business. No lab/office hours Papers Format- subheadings Progress Reports Early drafts – Thanksgiving Rachlin Summaries Tests Review Rewrites. Matching Continued. Herrnstein’s Hyperbola -- Application. Given this equation: What two ways could we reduce B?.

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Class Business

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  1. Class Business • No lab/office hours • Papers • Format- subheadings • Progress Reports • Early drafts – Thanksgiving • Rachlin • Summaries • Tests • Review • Rewrites

  2. Matching Continued

  3. Herrnstein’s Hyperbola -- Application Given this equation: What two ways could we reduce B?

  4. K = 10, R1=10, Re=10 10*10/(10+10) = 5

  5. Decrease Rf for the Behavior K = 10, R1=5, Re=10 10*5/(5+10) = 3.33

  6. Increase Re K = 10, R1=10, Re=20 10*10/(10+20) = 3.33 We do this all the time…

  7. Herrnstein's law and concurrent schedules The strict-matching equations for describing rates of responding during either alternative in concurrent VI VI schedules are: Alternative 1: Alternative 2:

  8. According to Herrnstein's assumptions, the denominators (B1 + B2 + Be) both equal k, so: and If now we divide B1 by B2 to see what relative behaviour allocation would look like, we get:

  9. As we saw already, this is just the same relation as: which is the strict matching law. So, the response rate on one of the schedules comprising a concurrent VI VI schedule is:

  10. How good is Herrnstein's law (1)? A law is only as good as the assumptions are true. It fits the data well, But is that enough? -- many other equations could be good fits.

  11. How good is Herrnstein's law (2)? The assumption that total output (k) is constant seems counter-intuitive, but it could be correct if you measure all the responses in the same modulus -- which Herrnstein's theory does. Even doing nothing is behaving.

  12. How good is Herrnstein's law (3)? The assumption about Re existing -- that there are other reinforcers available -- cannot be faulted. Logically, it seems that this must be correct. BUT the assumption that Re remains constant when R1 is varied is probably unreasonable. It seems unlikely that Re could remain constant when Be varies considerably. Re surely must fall when R1 is increased.

  13. How good is Herrnstein's law (4)? The assumption of strict matching is likely to be wrong. Considerable research has shown that the behavior ratio changes rather less with changes in reinforcer ratios than is suggested by strict matching.

  14. GeneralizedMatching

  15. Strict matching law • This one is the relative form, because it shows relative response and reinforcer rates (proportions) : • You can rearrange this to give the ratio form : • Lots of concurrent-schedule experiments supported the strict matching law, at least roughly, but not all of them

  16. Staddon (1968) was the first to collect data that clearly didn’t fit the strict matching law He studied concurrent DRL VI(DRL) schedules in pigeons, and varied the VI schedule His data did NOT fit on the major diagonal on relative coordinates as required by the strict matching law....

  17. Strict matching is still shown as a diagonal line at 45º Equal reinforcers means log (R1/R2) = 0 Equal responses means log (B1/B2) = 0 The data clearly fall along a line, whose equation is y = .59x + .51 So Staddon (1968) plotted his data in log-ratio form, log(B1/B2) against log(R1/R2) :

  18. Two deviations from strict matching are visible in Staddon’s data : Undermatching – there’s less change in log response ratio than predicted by strict matching (the slope of the line is less than 1) Bias – over all log reinforcer ratios, there are more B1 responses than predicted by strict matching

  19. More undermatching and bias • At first, researchers didn’t take much notice of Staddon’s results – weird procedure, weird results • People like Herrnstein’s strict matching theory • simple and parsimonious • Reports of these deviations from strict matching continue • in more standard concurrent schedules • Hollard and Davison (1971) – concurrent VI VI, pigeons, food vs electrical brain stimulation as reinforcers • Trevett, Davison and Williams (1972) – concurrent VI FI, pigeons, food reinforcers • Baum (1974): concluded that undermatching was actually more common than strict matching

  20. The Generalized Matching Law - GML • Baum (1974) therefore suggested a more general form of the matching law that included parameters to describe undermatching and bias : • The GML has the form of the equation for a straight line : y = slope (x) + intercept • The slope a is called sensitivity to reinforcer rate (Lobb & Davison, 1975) • The intercept log c is called bias (Baum, 1974)

  21. The Generalized Matching Law - GML • Why is the equation in log-ratio form? Because otherwise it wouldn’t be a straight line. If we take the GML out of logs : becomes a power function : • Not a straight line, harder to interpret on a graph, and much harder to find the best-fitting values of a and log c

  22. The Generalized Matching Law • If sensitivity (a) < 1, undermatching • If sensitivity (a) > 1, overmatching • If bias (log c) > 0, biased towards Alternative 1 • If bias (log c) < 0, biased towards Alternative 2 • If a = 1 and log c = 0, strict matching • If a = 1 and log c 0, biased matching • For Staddon’s data, a = 0.59, so undermatching, and log c = 0.51, so biased towards Alternative 1 (the shorter DRL)

  23. Examples Three GML plots: Strict matching a = 1, log c = 0 Undermatching a < 1 , log c = 0 Biased undermatching a < 1 , log c < 0

  24. Examples Much harder to see what’s going on in relative plots

  25. Examples And even worse with ratio axes

  26. The Generalized Matching Law • Undermatching means that when we vary the log reinforcer ratio (the distribution of reinforcers between alternatives), the animal’s choice changes less than strict matching predicts • Undermatching is more common than strict matching – usual value of sensitivity is about 0.8 to 0.9 • Example – Alsop and Elliffe (1988) – six pigeons, food reinforcement, five different concurrent VI VI schedules

  27. Alsop and Elliffe (1988) The GML fitted the data well (data points close to line)

  28. Alsop and Elliffe (1988) All birds showed undermatching (slopes < 1)

  29. Alsop and Elliffe (1988) All showed some bias – 132 and 134 to B1, others to B2

  30. Response vs. time allocation • Just as with the strict matching law, we could measure choice in terms of time allocation instead of response allocation : • Results are very similar, but sensitivity is usually a little higher (a is closer to 1, less undermatching)

  31. Alsop and Elliffe (1988) All birds showed higher time than response sensitivities

  32. Why do we usually get undermatching? • Baum’s view – undermatching is a failure of matching • Organisms are ‘trying to match’ but various things prevent it – usually related to confusion between the alternatives • e.g., no COD or COD too short (Schroeder & Holland, 1969) – without a COD, reinforcers on one alternative will affect responses on the other, so not surprising that we get undermatching • e.g., stimuli signaling the alternatives may be confused • Miller, Saunders and Bourland (1980) ran a switching-key concurrent VI VI with the alternatives signaled by line orientations • Found almost strict matching when lines were 45º different, but a = 0.33 when lines were 15º different

  33. Why do we sometimes get overmatching? • It can’t just be a failure of matching though, because there are conditions that reliably produce overmatching (a > 1) • This happens if we make it difficult to switch between alternatives • e.g., Baum (1982) – two-key concurrent VI VI with pigeons, but they had to walk around a partition or jump over a hurdle to switch between keys – a was about 1.9 • e.g., Davison and Elliffe (2000) – switching-key concurrent VI VI, but a switch response produced a delay of 4.5 s before the other stimulus and schedule appeared – a was about 1.6 • This doesn’t seem consistent with Baum’s idea that strict matching is normative, but sometimes animals can’t do it

  34. Why do we get bias? • Much easier to understand – bias is a constant proportional preference for one alternative • doesn’t change when we change the reinforcer ratio • Choice is biased towards: • Large reinforcers over small reinforcers • Immediate reinforcers over delayed reinforcers • Easy responses over hard responses (e.g., force required to peck) • Higher quality reinforcers over lower quality reinforcers • Reinforcers that occur at unpredictable times (VI) over reinforcers that occur at regular times (FI) • There might also be some inherent bias specific to the individual subject, • like muscular differences: handedness (beakedness?)

  35. Why do we get bias? • All these variables affect choice. If one of them is constant but unequal on the two alternatives, it will produce a bias. • e.g., if left-key responses are reinforced with 3 s access to wheat, and right-key responses are reinforced with 6 s access to wheat, and we then vary the reinforcer rates on each key, we would expect the GML to fit the data well, but with a negative value of log c – a bias towards B2, the right key • e.g., Hollard and Davison (1971) – pigeons, left key always had 3 s food as the reinforcer, right key had 10 s of ectostriatal electrical brain stimulation, varied the reinforcer rate ratio over conditions

  36. Hollard and Davison (1971) • Sensitivity values look normal, but there was a big bias in favor of the alternative providing food reinforcers • Responses: bias to food = 0.65 in log units • 100.65 4.5, so birds liked food 4.5 times as much as EBS

  37. Hollard and Davison (1971) • Time: bias to food = 0.71 in log units • 100.71 5.1, so birds liked food 5.1 times as much as EBS • But there’s a problem with the design of this experiment …

  38. Hollard and Davison (1971) • We can’t be certain that the bias was because of the different reinforcer qualities, or because of inherent bias, or some other asymmetry between the keys • e.g., maybe the food key was easier to peck than the EBS key, and that’s why the birds were biased towards it • Hollard and Davison needed a series of control conditions where each key provided the same reinforcer, so they could measure any inherent bias etc and subtract it from the bias in the food v EBS conditions

  39. Trevett, Davison and Williams (1972) • Does GML applied to concurrent FI VI as well as to VI VI? • Is there a bias to either FI or VI? • Ran the control conditions • Bias should be due to a preference for FI or VI • Baseline (control): • Five concurrent VI VI schedules in which reinforcer ratio was varied from 1:4 to 4:1 (log reinforcer ratio varied from –0.6 to +0.6) • Experimental conditions: • Five concurrent FI VI schedules that varied reinforcer ratio over the same range

  40. Trevett, Davison and Williams (1972) • Sensitivities (slopes) not significantly different for the two schedule types • conc VI VI – small bias towards Key 2 (inherent bias? easier to peck?) • conc FI VI – bigger bias towards VI. Difference between the biases, 0.15 log units, must measure preference for VI over FI • Pigeons seem to prefer unpredictable reinforcers over predictable ones – risk-prone rather than risk-averse, perhaps • Similar results for time allocation

  41. White & Davison (1973) FI FI • White and Davison varied relative reinforcer rates in conc FI FI • Sometimes, the pigeons showed normal FI scallops, but sometimes the cumulative records looked like VI schedules (high constant response rate) • When they showed the same pattern of responses on both keys, sensitivity was close to 1 (strict matching) and there was no bias • When they showed different patterns on each key (one FI-like, one VI-like), they found undermatching, and a bias towards the VI-like alternative

  42. Concurrent VI VR • The results depend on whether you measure response or time allocation • Response bias towards VR schedule • Time bias towards VI schedule • Seems reasonable? • VR is response-based, VI is time-based • So response and time measures aren’t always the same, and can be in opposite directions • Davison (1982): strange result that is a problem for the GML: • Keep the VI schedule constant and vary the VR, get strict matching (a = 1) • Keep the VR constant and vary the VI, get undermatching. • So maybe the GML doesn’t apply to ratio schedules very well.

  43. Other IVs in the GML • Reinforcer rates control choice according to the GML • Other independent variables, like reinforcer magnitudes, delays, qualities, etc. produce biases in the GML if they are held constant but unequal while we vary reinforcer rates • What if we keep the reinforcer rates constant and vary something else instead? • e.g., arrange conc VI 60 s VI 60 s and vary magnitude

  44. Schneider (1973) and Todorov (1973) • Version of the GML that describes the effect of reinforcer magnitude on choice • Schneider and Todorov both varied reinforcer magnitude and found that the above equation did describe their data • But sensitivity to magnitude was about 0.5 – much less than sensitivity to rate • Reinforcer rate is more effective at influencing choice than reinforcer magnitude

  45. The concatenated generalized matching law • If we put the GML descriptions of control by rate and magnitude together, we could write the above equation • The sensitivity terms have subscripts identifying the IV they refer to • Each IV controls choice in the same way, but sensitivity to each might differ • Because reinforcer rate affects choice more than reinforcer magnitude does, ar is greater than am

  46. The concatenated generalized matching law • We could just keep going, adding a log ratio and a sensitivity term for all the other IVs that affect choice • Notice that reinforcer delay and response arduousness have reversed log ratios – why? • Sensitivity term: how much influence that IV has on choice • If an IV is constant and equal for both alternatives, its log ratio will be zero and it will drop out of the equation • If an IV is constant and unequal it will produce a bias

  47. The concatenated generalized matching law • e.g., suppose M1 is 6 s access to wheat and M2 is 3 s • We know that am is about 0.5 • The magnitude ratio is 2, so log magnitude ratio is 0.3, so the magnitude term is about 0.5 x 0.3 = 0.15 • So if we vary the reinforcer rates and measure bias, it should be about 0.15 towards the larger magnitude, as long as all the other IVs in the concatenated GML are constant and equal, and there’s no inherent bias

  48. Vollmer and Bouret (2000) American college basketballers, 3-point shots vs 2-point shots Across players, ratio of shots attempted (responses) slightly undermatched ratio of successful shots (reinforcer rate) with a bias in favour of 3-point shots (reinforcer magnitude)

  49. The concatenated generalized matching law • This is an efficient way to write the concatenated GML. The symbol S means “the sum of” • X means any independent variable that affects choice • The equation says that the predicted log response ratio is the sum of all the log independent-variable ratios multiplied by their sensitivities • There’s no bias term in the equation, because the hope is that eventually, when all the IVs that affect choice have been discovered, there will be no inherent bias

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