Decision Making In Prisoner’s Dilemma

8.5 Variables

The independent variable is, essentially, the length of the game – or the probability that the opponent will be encountered in subsequent moves by the subject and may (or may not) reciprocate subject’s behavior from previous move or moves. In single-move Prisoner’s Dilemma the probability of another encounter is zero, in iterated Prisoner’s Dilemma it is considerably higher, although not always easily calculated (see section 6.19).

The dependent variable here is the frequency of cooperation.
(Note: the frequency of cooperation can be easily obtained from the amount of cooperation – it is the amount of cooperation for a certain number of moves, in our study typically for 10 moves. The reason why we use both measures is that frequency of cooperation enables comparison among games of different length.)
Now, the exact measures of cooperation (the dependent variables) that will be compared to test the hypotheses in this section are:
Variable “36”: The frequency of cooperative (C) choices played by the subject in moves 1-30 (for 10 moves)
Variable “37”: The frequency of C choices played by the subject in moves 1-30 in all Tit for Tat conditions (for 10 moves)
Variable “38”: The frequency of C choices played by the subject in moves 1-30 in all Random conditions (for 10 moves)
Variable “39”: The frequency of C choices played by the subject in moves 1-30 in all Benevolent conditions (for 10 moves)
Variable “40”: The frequency of C choices played by the subject in moves 1-30 in all Deterrent conditions (for 10 moves)
Variable “41”: The frequency of C choices played by the subject in moves 1-30 in all Bully conditions (for 10 moves)
Variable “42”: The frequency of C choices played by the subject in moves 26-30 in all Closed conditions (for 10 moves)
Variable “44”: The frequency of C choices played by the subject in all single-move games (for 10 moves)
Variable “62”: The frequency of C choices played by the subject in the last move of all closed-end games (for 10 moves)
For better orientation, the variables are often represented in text just by their number in quotation marks (“1”, “2”, “3”, etc.) or by v1, v2, v3, etc. The reader might look-up the full description in the respective Variables section or in the overview of all variables in Appendix. The conditions (the five strategies, and open, closed, and semiclosed games) are described in section 8.44.

8.6 Hypotheses

█ (H1) The frequency of cooperation in single-move games will be lower than the averaged frequency of cooperation in all iterated Prisoner’s Dilemma games taken together.

Null hypothesis is rejected and H1 is supported if “36” > “44”.

█ (H2) The frequency of cooperation in single-move games will be lower than the averaged frequency of cooperation in the last five moves in all closed-end iterated Prisoner’s Dilemma games taken together.

Null hypothesis is rejected and H2 is supported if “42” > “44”.

█ (H3) The frequency of cooperation in single-move games will be higher than the averaged frequency of cooperation in the last move for all closed-end iterated Prisoner’s Dilemma games taken together.

Null hypothesis is rejected and H3 is supported if “44” > “62”.

█ (H4) The frequency of cooperation in single-move games will be higher than the frequency of cooperation averaged for every strategy-specific iterated game (i. e. games using Tit for Tat, Random, etc. taken separately).

Null hypothesis is rejected and H4 is supported if

“37” > “44”, and/or

“38” > “44”, and/or

“39” > “44”, and/or

“40” > “44”, and/or

“41” > “44”.

8.7 Results

Ad H1

To test hypothesis H1 we used dependent t-test for repeated measures, and confirmed that “36” > “44”, p < 0,001 (for more details see Table 8.4 and the boxplot – Graph 8.1 – below). That means we confirmed hypothesis H1. (To be more precise, we rejected the respective null hypothesis, i. e. that frequency of cooperation in single-move games is the same as the averaged frequency of cooperation in all iterated Prisoner’s Dilemma games taken together, but from now on we will, for simplicity’s sake, only state that we either confirmed, or refuted the alternative hypothesis, i. e. H1, H2, etc.)

Graph 8.1: Variables “36” and “44”

(Graph 8.1: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 8.4: Comparing variables “36” and “44”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“36”	7,30	0,65
“44”	3,65	3,24	43	7,822	42	0,000

Ad H2
To test hypothesis H2 we used dependent t-test for repeated measures, and confirmed that “42” > “44”, p < 0,01 (for more details see Table 8.5 and Graph 8.2). That means we confirmed hypothesis H2: The frequency of cooperation in single-move games will be lower than the averaged frequency of cooperation in the last five moves for all closed-end iterated Prisoner’s Dilemma games taken together.

Graph 8.2: Variables “42” and “44”

(Graph 8.2: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 8.5: Comparing variables “42” and “44”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“42”	5,03	1,24
“44”	3,65	3,24	43	2,950	42	0,005

Ad H3
Using dependent t-test for repeated measures we confirmed that “44” > “62”, p < 0,001 (for more details see Table 8.6 and Graph 8.3). That means we confirmed hypothesis H3: The frequency of cooperation in single-move games will be higher than the averaged frequency of cooperation in the last move for all closed-end iterated Prisoner’s Dilemma games taken together.

Graph 8.3: Variables “44” and “62”

(Graph 8.3: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 8.6: Comparing variables “44” and “62”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“44”	3,65	3,24
“62”	0,58	1,40	43	5,745543	42	0,000

Ad H4
Using dependent t-test for repeated measures we confirmed that:
“37” > “44”, p < 0,001,

“39” > “44”, p < 0,001,

“40” > “44”, p < 0,001,

“41” > “44”, p < 0,001.

For more details see Tables 8.7, 8.8, 8.9, 8.10 and Graphs 8.4, 8.5, 8.6, 8.7:

Graph 8.4: Variables “37” and “44”

(Graph 8.4: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 8.7: Comparing variables “37” and “44”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“37”	9,12	0,89
“44”	3,65	3,24	43	11,223	42	0,000

* * * * * *

Graph 8.5: Variables “39” and “44”

(Graph 8.5: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 8.8: Comparing variables “39” and “44”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“39”	6,86	1,75
“44”	3,65	3,24	43	6,695520	42	0,000

* * * * * *

Graph 8.6: Variables “40” and “44”

(Graph 8.6: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 8.9: Comparing variables “40” and “44”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“40”	9,28	0,53
“44”	3,65	3,24	43	11,692	42	0,000

* * * * * *

Graph 8.7: Variables “41” and “44”

(Graph 8.7: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 8.10: Comparing variables “41” and “44”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“41”	7,09	1,47
“44”	3,65	3,24	43	6,772	42	0,000

* * * * * *

We did not find statistically significant difference in mean when variables “38” and “44” were compared with t-test (for more details see Table 8.11 and Graph 8.8).

Graph 8.8: Variables “38” and “44”

(Graph 8.8: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 8.11: Comparing variables “41” and “44”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“41”	3,98	1,03
“44”	3,65	3,24	43	0,685002	42	0,497

That means we partially confirmed hypothesis H4: The frequency of cooperation in single-move games will be lower than the frequency of cooperation averaged for every strategy-specific iterated game.

To be more precise, the performed tests confirm hypothesis H4’: The frequency of cooperation in single-move games will be lower than the frequency of cooperation averaged for certain strategy-specific iterated games (i. e. for games, where Tit for Tat, Benevolent, Bully, and Deterrent strategy is used).