PostgreSQL define campo do objeto JSON no array JSON

CREATE TABLE test_table(
  id BIGSERIAL PRIMARY KEY ,
  game TEXT,
  players JSONB
);

INSERT INTO test_table(game, players)
    VALUES
      ('chess', '[{"name": "Joe", "role": "admin"}, {"name": "Mike", "role": "user"}]'),
      ('football', '[{"name": "Foo", "role": "user"}, {"name": "Bar", "role": "user"}]');

+----+----------+----------------------------------------------------------------------+
| id |   game   |                               players                                |
+----+----------+----------------------------------------------------------------------+
|  1 | chess    | [{"name": "Joe", "role": "admin"}, {"name": "Mike", "role": "user"}] |
|  2 | football | [{"name": "Foo", "role": "user"}, {"name": "Bar", "role": "user"}]   |
+----+----------+----------------------------------------------------------------------+

WITH json_rows AS
(SELECT id, jsonb_array_elements(players) as json_data FROM test_table
WHERE game = 'chess'),
 updated_rows AS (
    SELECT
      id,
      array_to_json(array_agg(
      CASE WHEN json_data -> 'name' = '"Joe"'
        THEN jsonb_set(json_data, '{role}', '"user"')
      ELSE json_data END)) as updated_json
    FROM json_rows
    GROUP BY id
)
UPDATE test_table SET players = u.updated_json
FROM updated_rows u
WHERE test_table.id = u.id;

+----+----------+---------------------------------------------------------------------+
| id |   game   |                               players                               |
+----+----------+---------------------------------------------------------------------+
|  2 | football | [{"name": "Foo", "role": "user"}, {"name": "Bar", "role": "user"}]  |
|  1 | chess    | [{"name": "Joe", "role": "user"}, {"name": "Mike", "role": "user"}] |
+----+----------+---------------------------------------------------------------------+

1. 
   Converta a matriz json para linhas json e filtre-as pelo game propriedade. Isso é feito criando o json_rows CTE.
   
2. 
   Atualize os dados json nas linhas json onde o usuário "Joe" é encontrado.
   
3. 
   Depois de ter os novos valores json, basta fazer uma atualização com base no id.