Decision Making In Prisoner’s Dilemma

9.2 Procedure

For testing hypotheses H5 – H9 we will use data obtained in the fifteen 30-move games played with each subjects, as described in section 8.44.

Data for testing hypotheses H10 and H11 are derived from a procedure specially designed for testing cheap talk effect on decision making (the procedure is described in section 11.2 in more detail). Here it suffices to say that players simply state their preliminary willingness to cooperate in moves 1-10, 11-20, and 21-30 in condition without cheap talk (condition X), as well as in condition with cheap talk (condition Y). They also state how much cooperation they expect from their opponent in moves 1-10, 11-20, and 21-30 in condition without cheap talk (X), and in condition with cheap talk (Y). For the purpose of testing hypotheses H10 and H11, only data obtained in condition (X) will be compared.

9.3 Variables

The independent variable is the “open-endedness”/“closed-endedness”/“semi-closed-endedness” of the games. Each subject plays five open-ended games, five games with fixed end (30 moves) and five games with vaguely closed end (25-35 moves). The order of encounters with these types of games is balanced among subjects. The conditions (the five strategies, and open, closed, and semiclosed games) are described in section 8.44.

The depended variable is the frequency of cooperation. The exact measures of cooperation that will be compared to test the hypotheses below are:
Variable “23”: The amount of C choices played by the subject in moves 26-30 in all closed conditions
Variable “24”: The amount of C choices played by the subject in moves 26-30 in all semiclosed conditions
Variable “25”: The amount of C choices played by the subject in moves 26-30 in all open conditions
Variable “26”: The amount of C choices played by the subject in moves 21-25 in all closed conditions
Variable “27”: The amount of C choices played by the subject in moves 21-25 in all semiclosed conditions
Variable “29”: The amount of C choices played by the subject in moves 21-30 in all closed conditions
Variable “30”: The amount of C choices played by the subject in moves 21-30 in all semiclosed conditions
Variable “32”: The amount of C choices played by the subject in moves 1-25 in all closed conditions
Variable “33”: The amount of C choices played by the subject in moves 1-25 in all semiclosed conditions
Variable “34”: The amount of C choices played by the subject in moves 1-25 in all open conditions
Variable “49”: The willingness to cooperate (the amount of C choices) stated by the subject in moves 1-10
Variable “50”: The willingness to cooperate (the amount of C choices) stated by the subject in moves 11-20
Variable “51”: The willingness to cooperate (the amount of C choices) stated by the subject in moves 21-30
Variable “52”: The willingness to cooperate (the amount of C choices) on the opponent’s part expected by the subject in moves 1-10
Variable “53”: The willingness to cooperate (the amount of C choices) on the opponent’s part expected by the subject in moves 11-20
Variable “54”: The willingness to cooperate (the amount of C choices) on the opponent’s part expected by the subject in moves 21-30

9.4 Hypotheses

█ (H5) The frequency (amount) of cooperation preceding the 30th move (that is in moves 25-30) will be highest in games with no fixed end.

Null hypothesis is rejected and H5 is supported if:

“25” > “23” and

“25” > “24”.

█ (H6) The frequency (amount) of cooperation in moves 25-30 will be lowest in the closed-end games.

Null hypothesis is rejected and H6 is supported if:

“25” > “23” and

“24” > “23”.

█ (H7) The frequency (amount) of cooperation will decline sooner in semiclosed games than in closed-end games, which can be operationalized in the folloing way: the frequency of C will be lower in semiclosed games in moves 20-25 than in the closed-end games.

Null hypothesis is rejected and H7 is supported if “26” > “27”.

█ (H8) The overall decline of cooperation will be more pronounced in closed-end games than in semiclosed games, that is the frequency of cooperation in closed-end games in moves 20-30 will be lower than the frequency of cooperation in 20th-30th moves in v semiclosed games.

Null hypothesis is rejected and H8 is supported if “30” > “29”.

█ (H9) The overall frequency of cooperation during the game (not including the end of the game), which will be operationalized as the frequency of C in moves 1-25, will be the same for all conditions (closed, semiclosed, and open games).

Null hypothesis is rejected and H9 is supported if:
“32” = “33” and

“32” = “34” and

“33” = “34”.

█ (H10) Willingness to cooperate (the amount of C choices) stated by subjects will decrease in the last ten moves of an iterated game.

Null hypothesis is rejected and H10 is supported if:

“49” > “51”, and/or

“50” > “51”.

█ (H11) Subjects’ expectation of their opponent’s willingness to cooperate (the amount of C choices) will decrease in the last ten moves of an iterated game.

Null hypothesis is rejected and H11 is supported if:

“52” > “54”, and/or

“53” > “54”.

9.5 Results

Ad H5

With dependent t-test for repeated measures we confirmed that “25” > “23”, p < 0,001 (for more details see Table 9.1 and Graph 9.1).

Graph 9.1: Variables “25” and “23”

(Graph 9.1: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 9.1: Comparing variables “25” and “23”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“25”	18,09	2,94
“23”	12,51	3,05	45	8,631	44	0,000

With dependent t-test for repeated measures we did not confirm that “25” > “24” (i. e. the null hypothesis was not falsified, for more details see Table 9.2 and Graph 9.2).

Graph 9.2: Variables “25” and “24”

(Graph 9.2: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 9.2: Comparing variables “25” and “24”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“25”	18,09	2,94
“24”	17,09	2,73	45	1,748	44	0,087

That means we partially confirmed hypothesis H5. More exactly, we confirmed that the amount of cooperation is higher in open games when compared to the amount of cooperation in closed games, but not in semiclosed games. This, nevertheless, still partially confirms the existence of an end-game effect.

Ad H6
With dependent t-test for repeated measures we confirmed that “25” > “23”, p < 0,001, and “24” > “23”, p < 0,001 (for more details see Table 9.1 and Graph 9.1 above and Table 9.3 and Graph 9.3 below).
Graph 9.3: Variables “24” and “23”

(Graph 9.3: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 9.3: Comparing variables “25” and “24”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“24”	17,09	2,73
“23”	12,51	3,05	45	8,631	44	0,000

That means we partially confirmed hypothesis H6 and the existence of an end-game effect.

Ad H7
To test hypothesis H7 we used dependent t-test for repeated measures. We found no significant difference between “26” and “27” (for more details see Table 9.4 and Graph 9.4).

Graph 9.4: Variables “26” and “27”

(Graph 9.4: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 9.4: Comparing variables “26” and “27”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“26”	17,80	2,81
“27”	17,96	2,92	45	-0,307	44	0,761

Additionally, we also tested, whether “28” > “26” and “28” > “27”. With dependent t-test for repeated measures we found no statistically significant difference in these two pairs of variables (for more details see Tables 9.5 and 9.6).

Table 9.5: Comparing variables “26” and “28”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“26”	17,80	2,81
“28”	18,18	2,98	45	-0,654	44	0,516487

Table 9.6: Comparing variables “27” and “28”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“27”	17,96	2,92
“28”	18,18	2,98	45	-0,353	44	0,72562

Hypothesis H7 was not confirmed. This result and the additional tests (see Table 9.5 and 9.6) demonstrate that the amount of cooperation in moves 20-25 is the same in closed, semiclosed, and open games.

Ad H8
With dependent t-test for repeated measures we confirmed that “30” > “29”; p < 0,001 (for more details see Table 9.7 and Graph 9.5 below).
Graph 9.5: Variables “29” and “30”

(Graph 9.5: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 9.7: Comparing variables “29” and “30”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“29”	30,31	5,50
“30”	35,09	5,29	45	-5,009	44	0,000

We confirmed hypothesis H8: The overall decline of cooperation is more pronounced in closed-end games than in vaguely closed games, that is the frequency of cooperation in closed-end games in moves 20-30 is lower than the frequency of cooperation in 20th-30th moves in semiclosed games.

Ad H9
With dependent t-test for repeated measures we did not find statistically significant differences between “32” and “33”, “32” and “34”, and “33” and “34” (for details see Tables 9.8, 9.9, 9.10 and Graphs 9.6, 9.7, 9.8 below).
Graph 9.6: Variables “32” and “33”

(Graph 9.6: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 9.8: Comparing variables “32” and “33”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“32”	91,51	13,79
“33”	91,76	13,27	45	-0,109	44	0,914

* * * * * *

Graph 9.7: Variables “32” and “34”

(Graph 9.7: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 9.9: Comparing variables “32” and “34”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“32”	91,51	13,79
“34”	93,36	11,49	45	-0,871540	44	0,388

* * * * * *

Graph 9.8: Variables “33” and “34”

(Graph 9.7: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices)

Table 9.10: Comparing variables “33” and “34”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“33”	91,76	13,27
“34”	93,36	11,49	45	-0,626	44	0,534

The result is that hypothesis H9 was supported (in this case H9 would be refuted, if statistically significant differences between the respective variables were found). In other words, the overall frequency of cooperation during the game (not including the end of the game), which was operationalized as the frequency of C choices in moves 1-25, was the same in all conditions (closed, semiclosed, and open games).

Ad H10
With dependent t-test for repeated measures we found that “49” > “51”, p < 0,001, we also found that “50” > “51”, p < 0,05 (see Tables 9.11 and 9.12 and Graphs 9.9 and 9.10).
Graph 9.9: Variables “49” and “51”

(Graph 9.9: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of subject’s own expected C choices)

Table 9.11: Comparing variables “49” and “51”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“49”	7,24	1,00
“51”	6,58	1,34	45	3,96412	44	0,000

* * * * * *

Graph 9.10: Variables “50” and “51”

(Graph 9.10: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of subject’s own expected C choices)

Table 9.12: Comparing variables “50” and “51”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“50”	7,00000	1,20604
“51”	6,58	1,34	45	2,68456	44	0,010

That means hypothesis H10 was confirmed. The difference in willingness to cooperate when moves 1-10 and 21-30 are compared is bigger in comparison with the difference between moves 11-20 and 21-30, as might be expected. Both differences are statistically significant.

Ad H11
With dependent t-test for repeated measures we found that “52” > “54”, p < 0,05 and

“53” > “54”, p < 0,05 (see Tables 9.13 and 9.14 and Graphs 9.11 and 9.12).

Graph 9.11: Variables “52” and “54”

(Graph 9.11: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices expected by subject on his opponent’s part)

Table 9.13: Comparing variables “52” and “54”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“52”	6,76	0,83
“54”	6,36	1,11	45	2,362	44	0,023

* * * * * *

Graph 9.12: Variables “53” and “54”

(Graph 9.12: □ = mean, box = mean ± SD, whiskers = mean ± 1,96 SD; y axis shows number of C choices expected by subject on his opponent’s part)

Table 9.14: Comparing variables “53” and “54”

Variables	Dependent t-test for repeated measures
	Mean	SD	N	t	df	p
“53”	6,67	0,95
“54”	6,36	1,11	45	2,256	44	0,029

That means hypothesis H11 was confirmed and the existence of an end-game effect was, again, supported.